MAFELAP 2013

Conference on the Mathematics of Finite Elements and Applications 10–14 June 2013

Abstracts MAFELAP 2013

The organisers of MAFELAP 2013 are pleased to acknowledge the nancial support given to the conference by the Institute of Mathematics and its Applications (IMA) in the form of IMA Stu- dentships. Contents of the MAFELAP 2013 Abstracts Invited, parallel and mini-symposium order

Invited talks Finite Element Methods in Coastal Ocean Modeling: Successes and Challenges Clint N Dawson ZIENKIEWICZ LECTURE ...... 1-1 Forty years of the Crouzeix-Raviart element Susanne C. Brenner BABUSKAˇ LECTURE ...... 1-1 Advances in reducing the mesh burden in computational mechanics applications to fracture and surgical simulation St´ephaneP.A. Bordas ...... 1-2

Fields, control fields, and commuting diagrams in isogeometric analysis Annalisa Buffa, Giancarlo Sangalli and Rafael V´azquez ...... 1-4

The inf-sup constant of the divergence Martin Costabel ...... 1-5

Finite element methods for surface PDEs Charles M. Elliott ...... 1-6

A stochastic collocation approach to PDE-constrained optimization for random data identification problems Max Gunzburger ...... 1-7

Time Domain Integral Equations for Computational Electromagnetism Peter Monk ...... 1-8

The Emergence of Predictive Computational Science: Validation and Verification of Computational Models of Complex Physical Systems J. Tinsley Oden ...... 1-9

What is the Largest Finite Element System that can be Solved Today? Ulrich Ruede ...... 1-10

i Double complexes and local bounded cochain projections Ragnar Winther ...... 1-10

Talks in parallel sessions

Verification of functional a posteriori error estimates for obstacle problem Petr Harasim and Jan Valdman ...... 2-2 Convergence of hp-FEM in Three Dimensional Computation of Thermoelectric Effects Razi Abdul-Rahman and Sarah Kamaludin ...... 2-3 Discontinuous Galerkin time stepping schemes combined with local projection stabi- lization methods applied to transient Stokes problems: stability and convergence Naveed Ahmed and Gunar Matthies ...... 2-4 Anisotropic mesh adaptation for the Ambrosio-Tortorelli model: application to quasi- static crack propagation Marco Artina, Massimo Fonrasier, Simona Perotto and Stefano Micheletti ...... 2-5 A posteriori error analysis for time-dependent Stokes equations Eberhard B¨ansch, Johan Jansson, Fotini Karakatsani and Charalambos Makri- dakis ...... 2-6 Non-Conforming Finite Element Methods for the Obstacle Problem Carsten Carstensen and Karoline K¨ohler ...... 2-7 On the numerical simulation of well stability Philippe R B Devloo, Erick Raggio Slis Santos and Diogo Lira Cec´ılio ...... 2-8 Estimation of discretization and algebraic error via quasi-equilibrated fluxes for dis- continuous Galerkin methods V´ıtDolejˇs´ı,Ivana Sebestov´aˇ and Martin Vohral´ık ...... 2-9 Condensing the spectral element method for time domain wave problems Dugald B Duncan and Mark Payne ...... 2-10 Local mass conservation of Stokes finite elements Daniele Boffi, Nicola Cavallini, Francesca Gardini and Lucia Gastaldi ...... 2-11 Enriching a Hankel Basis by Ray Tracing in the Ultra Weak Variational Formulation C. J. Howarth, Simon Chandler-Wilde, Stephen Langdon and P.N. Childs...... 2-12

ii Error estimates for nonlinear convective and singularly perturbed problems in finite element methods V´aclav Kuˇcera ...... 2-14 A domain decomposition method with an optimized penalty parameter Chang-Ock Lee and Eun-Hee Park ...... 2-15 The Partition of Unity Method for the 3D elastic wave problems in the high frequency domain M. Mahmood, O. Laghrouche, A. El-Kacimi and J. Trevelyan ...... 2-16 On the numerical treatment of essential boundary conditions within positivity-preserving finite element methods for convection-dominated transport problems Matthias M¨oller ...... 2-18 Algebraic Flux Correction in a Partial Differential-Algebraic Framework Julia Niemeyer and Bernd Simeon ...... 2-19 Approximation of eddy currents in an axisymmetric unbounded domain Pilar Salgado and Virginia Selgas ...... 2-21 A uniform convergence analysis of three-step Taylor Galerkin finite element monotone iterative domain-decomposition scheme for singularly perturbed problems Vivek Sangwan and B. V. Rathish Kumar ...... 2-22 With a hierarchical error indicator toward anisotropic mesh refinement Ren´eSchneider ...... 2-23 Computational Aspects in Smooth Approximation of Data Karel Segeth ...... 2-23

iii Talks in Mini-Symposium

A priori finite element error estimates in optimal control A priori error estimates for finite element methods for H(2,1)-elliptic equations Thomas Apel, Thomas G. Flaig and Serge Nicaise ...... 3-2

Crank-Nicolson and St¨ormer-Verlet discretization schemes for optimal control problems with parabolic partial differential equations Thomas Apel and Thomas G. Flaig ...... 3-3

Error estimates for Dirichlet control problems in polygonal domains Thomas Apel, Mariano Mateos, Johannes Pfefferer and Arnd R¨osch ...... 3-4

Boundary concentrated FEM for optimal control problems Sven Beuchler ...... 3-5

Error estimates for the velocity tracking problem using duality arguments Konstantinos Chrysafinos ...... 3-5

Convergence and error analysis of a numerical method for the identification of matrix parameters in elliptic PDEs Klaus Deckelnick and Michael Hinze ...... 3-6

Optimal Control of Biharmonic Operator Stefan Frei, Rolf Rannacher and Winnifried Wollner ...... 3-7

An interior penalty method for distributed optimal control problems governed by the biharmonic operator Thirupathi Gudi, Neela Nataraj and Veeranjaneyulu Sadhanala ...... 3-7

A priori error estimates for parabolic optimal control problems with point controls Dmitriy Leykekhman and Boris Vexler ...... 3-8

Optimal error estimates for finite element discretization of elliptic optimal control prob- lems with finitely many pointwise state constraints Dmitriy Leykekhman, Dominik Meidner and Boris Vexler ...... 3-9

iv Verification of optimality conditions and discretization error estimates Martin Naß and Arnd R¨osch ...... 3-10

On discretized nonconvex elliptic optimal control problems with pointwise state con- straints Ira Neitzel, Johannes Pfefferer and Arnd R¨osch ...... 3-10

Sparse Elliptic Control Problems in Measure Spaces: Regularity and FEM Discretiza- tion Konstantin Pieper and Boris Vexler ...... 3-11

Optimal boundary control problems in energy spaces Olaf Steinbach ...... 3-11

Finite element methods for fourth order variational inequalities arising from elliptic optimal control problems Li-yeng Sung ...... 3-12

Analysis and applications of boundary element methods The BEM++ boundary element library and applications S.R. Arridge, T. Betcke, M. Schweiger and W. Smigaj´ ...... 4-2

A recursive integral equations approach for electromagnetic scattering by biperiodic multilayer gratings Beatrice Bugert and Gunther Schmidt ...... 4-3

Black-Box Preconditioning of FEM/BEM matrices by H-matrix techniques Markus Faustmann, Jens Markus Melenk and Dirk Praetorius ...... 4-4

One-equation FEM-BEM coupling for elasticity problems Michael Feischl, Thomas F¨uhrer, Michael Karkulik and Dirk Praetorius ...... 4-5

An axiomatic approach to optimality of adaptive algorithms with applications to BEM Michael Feischl and Dirk Praetorius ...... 4-7

v A reduced basis boundary element method for a class of parameterized electromagnetic scattering model M. Ganesh, J. S. Hesthaven and B. Stamm ...... 4-8

Retarded potential boundary integral equations for sound radiation in a half-space Heiko Gimperlein ...... 4-9

Analysis of a non-symmetric coupling of Interior Penalty DG and BEM Norbert Heuer and Francisco-Javier Sayas ...... 4-9

Adaptive nonconforming boundary element methods Norbert Heuer and Michael Karkulik ...... 4-10

Parallel BEM-Based Methods Dalibor Luk´aˇs, Michal Merta, Luk´aˇsMal´y,Petr Kov´aˇrand Tereza Kov´aˇrov´a ...... 4-11

On the quasi-optimal convergence in FEM-BEM coupling Jens Markus Melenk, Dirk Praetorius and B. Wohlmuth ...... 4-12

On the Ellipticity of Coupled Finite Element and One-Equation Boundary Element Methods for Boundary Value Problems G¨unther Of and Olaf Steinbach ...... 4-13

Radial Basis Functions with Applications to Elasticity Sergej Rjasanow and Richards Grzhibovskis ...... 4-14

Stokes flow about a collection of slip solid particles A. Sellier ...... 4-15

Boundary Element Methods for Acoustic Resonance Problems Gerhard Unger ...... 4-16

BEM Based Shape Optimization Using Shape Calculus and Multiresolution Analysis Jan Zapletal, Kosala Bandara, Fehmi Cirak, G¨unther Of and Olaf Steinbach ...... 4-17

vi Boundary-Domain Integral Equations Localized boundary-domain integral equations approach for Dirichlet and Robin prob- lems of the theory of piezo-elasticity for inhomogeneous solids Otar Chkadua ...... 5-2

Numerics and spectral properties of boundary domain integral and integro-differential operators in 3D Richards Grzhibovskis and Sergey E. Mikhailov ...... 5-3

Spectral properties and perturbations of boundary-domain integral equations Sergey E. Mikhailov ...... 5-4

Acoustic scattering by inhomogeneous anisotropic obstacle: Boundary-domain integral equation approach David Natroshvili ...... 5-6

Computational Micromagnetics magnum.fe: A micromagnetic finite-element code based on FEniCS Claas Abert ...... 6-2

Multiscale simulation of magnetic nanostructures Florian Bruckner ...... 6-3

Finite element and boundary element method in magnetic spin transport and magnetic hybrid structures Gino Hrkac, Marcus Page and Dieter Suess ...... 6-4

Coupling and numerical integration of LLG Marcus Page and Dirk Praetorius ...... 6-5

A nonlocal parabolic and hyperbolic model for type-I superconductors Karel Van Bockstal and Marian Slodiˇcka ...... 6-6

vii Computational challenges in Discontinuous Galerkin methods Energy stability for discontinuous Galerkin approximation of a problem in elasotody- namics Paola F. Antonietti, Blanca Ayuso de Dios, Ilario Mazzieri and Alfio Quarteroni ...... 7-2

Staggered discontinuous Galerkin methods for Maxwell’s equations Eric Chung ...... 7-3

Efficient Discontinuous Galerkin method for meteorological applications Andreas Dedner and Robert Kl¨ofkorn ...... 7-4

Discontinuous Galerkin Methods for Phase Field Models of Moving Interface Problems Xiaobing Feng ...... 7-5

Discontinuous Galerkin methods for non-linear interface problems Emmanuil H. Georgoulis ...... 7-5

On the convergence of adaptive discontinuous Galerkin methods Thirupathi Gudi and Johnny Guzm´an ...... 7-6

A cochain complex for interior penalty methods: error estimates and multigrid through differential relations Guido Kanschat and Natasha Sharma ...... 7-7

Generalized DG-Methods for highly indefinite Helmholtz problems Jens Markus Melenk, Asieh Parsania and Stefan A. Sauter ...... 7-7

Convergence of High Order Methods for the Miscible Displacement Problem Beatrice Riviere ...... 7-8

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretiza- tions of advection-dominated flows Jaap van der Vegt and Sander Rhebergen ...... 7-9

Mixed hp-DGFEM for Linear Elasticity in 3D Thomas P. Wihler and Marcel Wirz ...... 7-10

viii Discontinuous Galerkin methods in fluid flows Hybridizable Discontinuous Galerkin Methods for the incompressible Oseen and Navier- Stokes equations Aycil Cesmelioglu, Bernardo Cockburn, Ngoc Cuong Nguyen and Jaime Peraire ...... 8-2

Commuting diagrams for the TNT elements on cubes Bernardo Cockburn and Weifeng Qiu ...... 8-2

Development and validation of a discontinuous Galerkin wave prediction model Ethan Kubatko and Angela Nappi ...... 8-3

Coupling of Stokes and Darcy Flows using Discontinuous Galerkin and Mimetic Finite Difference Method Konstantin Lipnikov, Danail Vassilev and Ivan Yotov ...... 8-4

Local Discontinuous Galerkin Method for Inkjet Drop Formation and Motion Tatyana Medvedeva, Onno Bokhove and Jaap van der Vegt ...... 8-5

Space-time (H)DG methods for incompressible flows Sander Rhebergen, Bernardo Cockburn and Jaap van der Vegt ...... 8-5

A Local Discontinuous Galerkin Method for the Propagation of Phase Transition in Solids Lulu Tian, Yan Xu, J.G.M. Kuerten and Jaap van der Vegt ...... 8-6

Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation Guaranteed lower bounds for eigenvalues Carsten Carstensen and Joscha Gedicke ...... 9-2

Adaptive path-following method for nonlinear PDE eigenvalue problems Carsten Carstensen, Joscha Gedicke, V. Mehrmann and Agnieszka Miedlar ...... 9-3

Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems Carsten Carstensen, Dietmar Gallistl and Mira Schedensack ...... 9-3

ix Computation of ground states of Schr¨odinger operator with large magnetic fields Monique Dauge ...... 9-4

Finite element analysis of a non-self-adjoint quadratic eigenvalue problem Christian Engstr¨om ...... 9-5

Solving an elliptic eigenvalue problem via automated multi-level sub-structuring and hierarchical matrices Peter Gerds and Lars Grasedyck ...... 9-6

Auxiliary subspace error estimation for high-order finite element eigenvalue approxi- mations Stefano Giani, Luka Grubiˇsi´c,Harri Hakula and Jeffrey S Ovall ...... 9-7

Kato’s square root theorem as a basis for relative estimation theory of eigenvalue approximations Stefano Giani, Luka Grubiˇsi´c, Agnieszka Miedlar and Jeffrey S Ovall ...... 9-8

High precision verified eigenvalue estimation for elliptic differential operator over polyg- onal domain of arbitrary shape Xuefeng Liu ...... 9-9

Spectral analysis for a mixed finite element formulation of the elasticity equations Salim Meddahi, David Mora and Rodolfo Rodr´ıguez ...... 9-10

Finite Elements for Elliptic Eigenvalue Problems in the Preasymptotic Regime Stefan A. Sauter ...... 9-10

Accurate Computations of Matrix Eigenvalues with Applications to Differential Oper- ators Qiang Ye ...... 9-11

Error Estimation and adaptive modelling Reduced basis finite element heterogeneous multiscale method for quasilinear problems Yun Bai ...... 10-2

x Error Estimation and Adaptive Modeling for Viscous Incompressible Flows Paul T. Bauman ...... 10-3

Goal-Oriented Error Estimation and Adaptivity for the Time-Dependent Low-Mach Navier-Stokes Equations Varis Carey and Paul T. Bauman ...... 10-3

Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs Alexandre Ern and Martin Vohral´ık ...... 10-4

Tree approximation versus AFEM Francesca Fierro, Alfred Schmidt and Andreas Veeser ...... 10-5

Contraction and Optimal Convergence of a Goal-Oriented Adaptive Finite Element Method Ricardo H. Nochetto, A.J. Salgado and K.G. van der Zee ...... 10-6

Finite Element Methods for Convection-Dominated Problems A computable error bound for a 3-dimensional convection-diffusion-reaction equation Mark Ainsworth, Alejandro Allendes, Gabriel R. Barrenechea and Richard Rankin

...... 11-2

Augmented Taylor-Hood Elements for Incompressible Flow Daniel Arndt ...... 11-3

A nonlinear dissipation to avoid local oscillations for the finite element approximation of the convection-diffusion equation Joan Baiges and Ramon Codina ...... 11-4

Investigations of a FEM-FCT scheme applied to a 1D model problem Gabriel R. Barrenechea, Volker John and Petr Knobloch ...... 11-5

A posteriori error estimation in stabilized discretizations of stationary convection- diffusion-reaction problems Markus Bause and Kristina Schwegler ...... 11-6

xi Robust error estimates in weak norms with application to implicit large eddy simulation Erik Burman ...... 11-7

Anisotropic Local Projection Stabilization in Streamline and Crosswind Directions Helene Dallmann and Gert Lube ...... 11-8

On Superconvergence for Higher-Order FEM in Convection-Diffusion Problems Sebastian Franz ...... 11-9

An adaptive SUPG method for evolutionary convection-diffusion equations Javier de Frutos, Bosco Garc´ıa-Archilla and Julia Novo ...... 11-9

SUPG finite element method for PDEs in time-dependent domains Sashikumaar Ganesan and Shweta Srivastava ...... 11-10

Stabilization of convection-diffusion problems by Shishkin mesh simulation Bosco Garc´ıa-Archilla ...... 11-11

A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations Volker John and Julia Novo ...... 11-11

Velocity-pressure reduced order models for the incompressible Navier–Stokes equations Volker John ...... 11-12

A Finite Element Method for a Noncoercive Elliptic Convection Diffusion Problem Klim Kavaliou and Lutz Tobiska ...... 11-12

On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime Alexander Linke ...... 11-13

A two-level local projection stabilisation on uniformly refined triangular meshes Gunar Matthies and Lutz Tobiska ...... 11-14

A Flux-Corrected Transport method based on local projection stabilization for non- stationary transport problems Friedhelm Schieweck and Dmitri Kuzmin ...... 11-15

xii Towards Anisotropic Quality Tetrahedral Mesh Generation Hang Si ...... 11-16

A local projection stabilization method for finite element approximation of a magne- tohydrodynamic model Benjamin Wacker and Gert Lube ...... 11-17

Finite Element Methods for Multiphysics Problems A stabilized finite volume element formulation for sedimentation-consolidation pro- cesses Raimund B¨urger, Ricardo Ruiz-Baier and H´ectorTorres ...... 12-2

Double layer potential boundary conditions for the Hybridizable Discontinuous Galerkin method Zhixing Fu, Norbert Heuer and Francisco-Javier Sayas ...... 12-3

A linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert equation Beniamin Goldys, Kim-Ngan Le and Thanh Tran ...... 12-4

A decoupled preconditioning technique for a mixed Stokes-Darcy model Antonio M´arquez, Salim Meddahi and Francisco-Javier Sayas ...... 12-5

Conforming and divergence-free Stokes elements Michael Neilan ...... 12-5

An exactly divergence-free finite element method for a generalized Boussinesq problem Ricardo Oyarz´ua and Dominik Sch¨otzau ...... 12-6 hp-Time-Discontinuous Galerkin for Pricing American Put Options Ernst P. Stephan ...... 12-6

Finite Elements in Nonlinear Spaces Subdivision Method for the Canhan-Helfrich model Jingmin Chen, Sara Grundel, Robert Kusner, Thomas Yu and Andrew Zigerelli ...... 13-2

xiii On Potts and Blake-Zisserman functionals for manifold-valued data Laurent Demaret, Martin Storath and Andreas Weinmann ...... 13-3

B-Spline quasiinterpolation of manifold-valued data Philipp Grohs ...... 13-4

Intrinsic discretization error bounds for geodesic finite element approximations of el- liptic minimization problems Hanne Hardering ...... 13-4

Simulation of Q-tensor fields with constant orientational order parameter in the theory of uniaxial nematic liquid crystals Alexander Raisch ...... 13-5

Finite elements for problems with singularities Eigenvalue problems in a non-Lipschitz domain Gabriel Acosta and Mar´ıaGabriela Armentano ...... 14-2

Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω) Thomas Apel, Ariel Lombardi and Max Winkler ...... 14-3

Strong convergence for Gauss’ law with edge elements Patrick Ciarlet, Haijun Wu and Jun Zou ...... 14-4 hp-Adaptive FEM Based on Continuous Sobolev Embeddings Thomas Fankhauser, Thomas P. Wihler and Marcel Wirz ...... 14-5

Mapping and regularity results for Schroedinger operators with inverse square poten- tials Eugenie Hunsicker, Hengguang Li, Victor Nistor and Vile Uski ...... 14-6

Finite Element Method for Schroedinger operators with inverse square potentials Eugenie Hunsicker, Hengguang Li, Victor Nistor, Jorge Sofo and Vile Uski ...... 14-7

Linear Finite Elements may be only First-Order Pointwise Accurate on Anisotropic Triangulations Natalia Kopteva ...... 14-8

xiv hp finite element methods for singularly perturbed transmission problems Serge Nicaise and Christos Xenophontos ...... 14-9

Foundations of isogeometric analysis Towards isogeometric analysis for compressible flow problems and unstructured meshes R. Abgrall ...... 15-2

Arbitrary-degree Analysis-suitable T-splines L. Beir˜aoda Veiga, Annalisa Buffa, Giancarlo Sangalli and Rafael V´azquez ...... 15-3

Isogeometric Analysis and Non-matching Domain Decomposition Methods Michel Bercovier ...... 15-5

Implementation of high order impedance boundary conditions in isogeometric methods Annalisa Buffa, Luca Di Rienzo and Rafael V´azquez ...... 15-6

A Computational Cost Analysis of Isogeometric Analysis Nathan Collier, Lisandro Dalcin, David Pardo, Maciej Paszynski and Victor Calo ...... 15-7

Mixed Isogeometric Collocation Methods for the Stokes Equations John A. Evans, Dominik Schillinger, Ren´eHiemstra and Thomas J.R. Hughes ...... 15-8

Algebraic Multilevel Preconditioning in Isogeometric Analysis Krishan Gahalaut and Satyendra Tomar ...... 15-9

Guaranteed and sharp a-posteriori error estimates in isogeometric analysis S.K. Kleiss and Satyendra Tomar ...... 15-10

Local refinements in IgA based on hierarchical generalized B-splines Carla Manni, Francesca Pelosi and Hendrik Speleers ...... 15-11

Efficient assembly method for isogeometric discretizations Angelos Mantzaflaris and Bert J¨uttler ...... 15-12

xv Comparison of boundary element method discretisation technologies for acoustic anal- ysis Robert N. Simpson, Michael A. Scott, Matthias Taus, Derek C. Thomas and Haojie Lian ...... 15-13

Splines on triangulations in isogeometric analysis Hendrik Speleers ...... 15-14

Approximation Properties of Singular Parametrizations in Isogeometric Analysis Thomas Takacs and Bert J¨uttler ...... 15-15

Adaptive Hierarchical B-Splines for Local Refinement in Isogeometric Analysis Anh-Vu Vuong and Bernd Simeon ...... 15-15

Global and local error estimates for problems with singularities or low regularity Error analysis of discontinuous Galerkin methods for the Stokes problem under minimal regularity Santiago Badia, Ramon Codina, Thirupathi Gudi and Johnny Guzm´an ...... 16-2

Optimality of an adaptive FEM for controlling local energy errors Alan Demlow ...... 16-2

Optimal error estimates for the parabolic problem in L∞(Ω; L2([0,T ])) norm Dmitriy Leykekhman and Boris Vexler ...... 16-3

A posteriori estimation of hierarchical type for a Schr¨odingeroperator with inverse square potential Hengguang Li and Jeffrey S Ovall ...... 16-4

Localized pointwise estimates for the fully nonlinear Monge-Amp`ereequation Michael Neilan ...... 16-5

Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm Francesca Tantardini, Andreas Veeser and R¨udigerVerf¨urth ...... 16-6

xvi High order finite element methods: A mini symposium cele- brating Leszek Demkowicz’s contributions FEM with discrete transparent boundary conditions for the Cauchy problem for the Schr¨odingerequation on the whole axis Alexander Zlotnik ...... 17-2

High Order FEM for Wave Propagation: Like it or lump it Mark Ainsworth ...... 17-3

Commuting Quasi interpolants for T-Spline Spaces Annalisa Buffa, Giancarlo Sangalli and Rafael V´azquez ...... 17-4

A PDE-constrained optimization approach to the discontinuous Petrov-Galerkin method with a trust region inexact Newton-CG solver Tan Bui-Thanh and Omar Ghattas ...... 17-5

A posteriori error control for DPG methods Carsten Carstensen, Leszek Demkowicz and Jay Gopalakrishnan ...... 17-6

Improved stability estimates for the hp-Raviart-Thomas projection operator on quadri- laterals Alexey Chernov and Herbert Egger ...... 17-7

DPG Method for Wave Propagation Problems, A Better Understanding Leszek Demkowicz, Jay Gopalakrishnan, Jens Markus Melenk, Ignacio Muga and David Pardo ...... 17-8

A space-time multigrid method for high order time discretizations Martin Gander, Martin Neum¨uller and Olaf Steinbach ...... 17-9

Partial expansion of a Lipschitz domain and some applications Jay Gopalakrishnan and Weifeng Qiu ...... 17-9

Dispersive and Dissipative Errors in the DPG Method with Scaled Norms for Helmholtz Equation Jay Gopalakrishnan, Ignacio Muga and Nicole Olivares ...... 17-10

xvii Adaptive and hybridized Hermite methods for initial-boundary value problems Thomas Hagstrom, Daniel Appel¨oand Ronald Chen ...... 17-11

On hp-Boundary Layer Sequences Harri Hakula ...... 17-12

Discontinuous Galerkin hp-BEM with quasi-uniform meshes Norbert Heuer and Salim Meddahi ...... 17-13

Two-Grid hp–Adaptive Discontinuous Galerkin Finite Element Methods for Second– Order Quasilinear Elliptic PDEs Paul Houston ...... 17-14

New Hybrid Discontinuous Galerkin Methods Youngmok Jeon and Eun-Jae Park ...... 17-15

Godunov SPH Methods for Simulating Complex Flows with Free Surfaces Over Rapidly Changing Natural Terrains Dinesh Kumar, E. B. Pitman and A. K. Patra ...... 17-16

Application of hp Finite Elements to the Accurate Computation of Polarisation Tensors for the Eddy Current Problem P.D. Ledger and W.R.B. Lionheart ...... 17-17

Recent advances in finite element simulation of electromagnetic wave propagation in metamaterials Jichun Li ...... 17-18 hp-FEM for singular perturbations with multiple scales Jens Markus Melenk and Christos Xenophontos ...... 17-19 hp Adaptive Finite Element Methods Based on Derivatives Recovery and Supercon- vergence Hieu Nguyen and Randolph E. Bank ...... 17-19

Comparison of different finite element models and methods for the Girkmann Shell- Ring Problem Antti H. Niemi and Julien Petit ...... 17-20

xviii Pyramidal finite elements Nilima Nigam, Argyrios Petras and Joel Phillips ...... 17-20

Application of the adaptive finite element method to numerical simulations of arteries Waldemar Rachowicz and Adam Zdunek ...... 17-21

Discontinuous Petrov-Galerkin Methods for Incompressible Flow: Stokes and Navier- Stokes Nathan V. Roberts, Leszek Demkowicz and Robert Moser ...... 17-22

Preconditioning for high order Hybrid DG Methods Joachim Sch¨oberl and Christoph Lehrenfeld ...... 17-23

Application of the fully automatic hp-FEM to elastic-plastic problems Marta Serafin and Witold Cecot ...... 17-24

A new error analysis for Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schr¨odingerequation Jilu Wang ...... 17-25

B-spline FEM approximation of wave equation Hongrui Wang and Mark Ainsworth ...... 17-25

The Low-storage Curvilinear Discontinuous Galerkin Method T. Warburton ...... 17-26

A novel formulation for nearly inextensible and nearly incompressible finite hyperelas- ticity Adam Zdunek, Waldemar Rachowicz and T. Eriksson ...... 17-27

Innovative compatible and mimetic discretizations for partial differential equations Basic Principles of Virtual Element Methods L. Beir˜aoda Veiga, Franco Brezzi, Andrea Cangiani, Gianmarco Manzini, L.D. Marini and Alessandro Russo ...... 18-2

xix Mimetic discretizations of elliptic problems Gianmarco Manzini ...... 18-3

A Virtual Element Method with high regularity L. Beir˜aoda Veiga and Gianmarco Manzini ...... 18-4

The Virtual Element Method for general second-order elliptic operators on polygonal and polyhedral meshes Franco Brezzi, L. Donatella Marini and Alessandro Russo ...... 18-5

Nonsmooth initial data error estimates for the finite volume element method for a parabolic problem Panagiotis Chatzipantelidis ...... 18-6

Convection dominated discontinuous Galerkin multiscale method Daniel Elfverson and Axel M˚alqvist ...... 18-7

MHM Method for Advective-Reactive Dominated Models Christopher Harder, Diego Paredes and Fr´ed´ericValentin ...... 18-8

Trefftz-DG methods for wave propagation: hp-version and exponential convergence Andrea Moiola, Ralf Hiptmair, Ilaria Perugia and Christoph Schwab ...... 18-9

The discrete maximum principle in the family of mimetic finite difference discretizations Daniil Svyatskiy, Konstantin Lipnikov and Gianmarco Manzini ...... 18-10

A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems Marco Verani ...... 18-11

Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics Norm-Optimal Iterative Learning Control for a Heating Rod Based on the Method of Integro-Differential Relations Harald Aschemann, Dominik Schindele and Andreas Rauh ...... 19-2

Variational Formulations of Inverse Dynamical Problems in Linear Elasticity Georgy Kostin and Vasily Saurin ...... 19-3

xx Design and Experimental Validation of Control Strategies for a Spatially Two-Dimensional Heat Transfer Process Based on the Method of Integro-Differential Relations Andreas Rauh, Luise Senkel and Harald Aschemann ...... 19-4

Integro-Differential Relations in Linear Elasticity: Static Case Vasily Saurin and Georgy Kostin ...... 19-6

Large scale computing with applications Adaptive Asynchronous Parallel Calculations at Petascale using Uintah Martin Berzins ...... 20-2

Accelerator-friendly parallel adaptive mesh refinement Carsten Burstedde, Lucas C. Wilcox, Georg Stadler and Donna Calhoun ...... 20-3

Total efficiency of core components in Finite Element frameworks Markus Geveler ...... 20-4

Massive parallel simulation of water and solute transport in porous media Olaf Ippisch, Markus Blatt and Jorrit Fahlke ...... 20-4

Scalable Parallel Multilevel Solution of Elliptic Problems Peter K. Jimack, Mark A. Walkley and Jianfei Zhang ...... 20-5

Thoughts on general purpose finite element libraries and hybrid programming Guido Kanschat ...... 20-6

Patching Adaptivity for Large Scale Problems - A New Lightweight Adaptive Scheme and its Application in Computational Electrocardiology Dorian Krause, Rolf Krause, Thomas Dickopf and Mark Potse ...... 20-6

Fast and Scalable Elliptic Solvers for Anisotropic Problems in Geophysical Modelling Eike Mueller, Robert Scheichl and Eero Vainikko ...... 20-7

Parallel Incompressible Flow Simulations using Divergence-Free Finite Elements Tobias Neckel ...... 20-8

xxi On Large-Scale Mechanics Simulations with the Parallel Toolbox Aurel Neic ...... 20-9

Recent Developments in NGSolve for Distributed and Many-Core Parallel Computing Joachim Sch¨oberl ...... 20-10

Analysis of adaptive space-time finite elements for parabolic problems Kunibert G. Siebert ...... 20-10

Scalable solvers for elliptic problems discretized by adaptive high-order finite elements Georg Stadler, Tobin Isaac, Hari Sundar, Carsten Burstedde and Omar Ghattas ...... 20-11

Algebraic multilevel preconditioning in H(curl) and H(div) space Satyendra Tomar ...... 20-11

Low Rank Tensor Based Numerical Methods Adaptive methods based on tensor representations of coefficient sequences and their complexity analysis Markus Bachmayr and Wolfgang Dahmen ...... 21-2

Black Box Approximation Strategies in the Hierarchical Tensor Format Jonas Ballani and Lars Grasedyck ...... 21-3

Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems Sergey V. Dolgov and Dmitry V. Savostyanov ...... 21-4 hp-DG time stepping for high-dimensional evolution problems with low-rank tensor structure Vladimir Kazeev ...... 21-5

Hartree-Fock eigenvalue solver using tensor-structured two-electron integrals Venera Khoromskaia ...... 21-6

Super-fast solvers for PDEs discretized in the quantized tensor spaces Boris Khoromskij ...... 21-7

xxii Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems Dmitry V. Savostyanov and Sergey V. Dolgov ...... 21-8

Mathematical and statistical modeling in biology Acoustic Localisation of Coronary Artery Stenosis: Wave Propagation in Soft Tissue Mimicking Gel H. Thomas Banks, Malcolm J. Birch, Mark P. Brewin, Steve E. Greenwald, Shuhua Hu, Zackary Kenz, Carola Kruse, Dwij Mehta, Simon Shaw and John R. Whiteman ...... 22-2

Efficient numerical methods for coupled PDE-ODE systems: An application in inter- cellular signaling Thomas Carraro, Elfriede Friedmann and Daniel Gerecht ...... 22-3

Modeling and inverse problem considerations for a viscoelastic tissue model Zackary Kenz ...... 22-3

High Order Space-Time Finite Element Schemes for the Dynamics of Viscoelastic Soft Tissue Carola Kruse, Simon Shaw, John R. Whiteman, H. Thomas Banks, Zackary Kenz, Shuhua Hu, Steve E. Greenwald, Mark P. Brewin and Malcolm J. Birch ...... 22-4

New advances in a posteriori error estimation Computable error bounds for finite element approximation on non-polygonal domains Mark Ainsworth and Richard Rankin ...... 23-2

Guaranteed and robust error bounds for singularly perturbed problems in arbitrary dimension Mark Ainsworth and Tom´aˇsVejchodsk´y ...... 23-3

Instance optimality for the maximum strategy Lars Diening ...... 23-4

A framework for robust a posteriori error control in unsteady nonlinear advection- diffusion problems V´ıtDolejˇs´ı,Alexandre Ern and Martin Vohral´ık ...... 23-5

xxiii Quasi-optimal AFEM for non-symmetric operators Michael Feischl, Thomas F¨uhrerand Dirk Praetorius ...... 23-6

A posteriori error estimates for the wave equation Omar Lakkis, Emmanuil H. Georgoulis and Charalambos Makridakis ...... 23-7

On Mathematical Methods Generating Fully Reliable A Posteriori Estimates for Non- linear Boundary Value Problems Sergey Repin ...... 23-8

Adaptive finite elements for PDE constrained optimal control problems Kunibert G. Siebert ...... 23-8

Non-Standard Finite Elements and Solvers in Solid Mechanics A new coarse space for FETI-DP in the context of almost incompressible elasticity Sabrina Gippert, Axel Klawonn and Oliver Rheinbach ...... 24-2

Nonlinear FETI-DP and BDDC Methods Axel Klawonn, Martin Lanser and Oliver Rheinbach ...... 24-2

LSFEM for geometrically and physically nonlinear elasticity problems Benjamin M¨uller, Gerhard Starke, J¨orgSchr¨oder,Alexander Schwarz and Karl Steeger

...... 24-3

An Approach to Adaptive Coarse Spaces in FETI-DP Methods Oliver Rheinbach, Axel Klawonn and Patrick Radtke ...... 24-3

Geodesic Finite Elements Oliver Sander ...... 24-4

Aspects on mixed least-squares finite elements for hyperelastic problems Alexander Schwarz, Karl Steeger, J¨orgSchr¨oder,Gerhard Starke and Benjamin M¨uller

...... 24-5

On isogeometric finite elements in solid mechanics and vibrational analysis Bernd Simeon and Oliver Weeger ...... 24-6

xxiv Momentum Balance in First-Order System Finite Element Methods for Elasticity Gerhard Starke ...... 24-7

Novel Methods for Time-Harmonic Wave Equations Analysis of a Cartesian PML approximation to acoustic scattering problems in Rn James H. Bramble and Joseph E. Pasciak ...... 25-2

A high frequency BEM for scattering by a class of nonconvex obstacles Simon Chandler-Wilde, David Hewett, Stephen Langdon and Ashley Twigger ...... 25-3

A high frequency boundary element method for scattering by two-dimensional screens Simon Chandler-Wilde, David Hewett, Stephen Langdon and Ashley Twigger ...... 25-4

Solving the steady-state ab-initio laser theory with FEM Sofi Esterhazy, Matthias Liertzer, Jens Markus Melenk and Stefan Rotter ...... 25-5

How should one choose the shift for the shifted Laplacian to be a good preconditioner for the Helmholtz equation? Martin Gander, I. G. Graham and E. A. Spence ...... 25-6

Hybrid numerical-asymptotic approximation for high frequency scattering by penetra- ble convex polygons Samuel Groth, David Hewett and Stephen Langdon ...... 25-7

Analysis of preconditoners for Helmholtz equation using Pesudospectrum Antti Hannukainen ...... 25-9

A domain decomposition preconditioner for mixed hybrid infinite elements Martin Huber, Lothar Nannen and Joachim Sch¨oberl ...... 25-10

Improving the Shifted Laplace Preconditioner by Multigrid Deflation A. H. Sheikh, D. Lahaye and C. Vuik ...... 25-11

xxv Numerical Methods for Parabolic Equations Energy conservative/dissipative approximations of nonlinear evolution problems Charalambos Makridakis ...... 26-2

A posteriori error analysis for dG in time ALE formulations Andrea Bonito, Irene Kyza and Ricardo H. Nochetto ...... 26-3

Discontinuous Galerkin Approximation of porous Fisher-Kolmogorov Equations Fausto Cavalli, Giovanni Naldi and Ilaria Perugia ...... 26-4

On adaptive discontinuous Galerkin methods for parabolic problems Emmanuil H. Georgoulis ...... 26-5

The hp-adaptive Galerkin time stepping method for nonlinear differential equations with finite time blow up B¨arbel Janssen and Thomas P. Wihler ...... 26-5

Maximum-norm strong approximation rates for noisy reaction-diffusion equations Omar Lakkis, G.T. Kossioris and M. Romito ...... 26-6

A new approach to error analysis of fully discrete finite element methods for nonlinear parabolic equations Buyang Li and Weiwei Sun ...... 26-7

Numerical Methods for Reaction-Transport Equations with Ap- plications in Medicine Finite element analysis of the mechano-chemical regulation of wound contraction in surgical wounds Etelvina Javierre, Clara Valero, Maria Jose Gomez-Benito and Jose Manuel Garcia- Aznar ...... 27-2

Presentation of results of finite-element analyses on a two-dimensional mechanochem- ical model for dermal wound healing D.C. Koppenol and Fred J. Vermolen ...... 27-3

xxvi Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell Anotida Madzvamuse, Uduak George and Angelique St´ephanou ...... 27-4

Analyzing the Treatment of a Bacterial Infection in a Wound Using Oxygen Therapy Richard Schugart ...... 27-5

A Semi–Stochastic Model for the Immune Response System Fred J. Vermolen ...... 27-6

Multiscale models of tumor cells: from in-vitro aggregates to in-vivo vascularized tu- mors Irene Vignon-Clementel, Nick Jagiella and Dirk Drasdo ...... 27-7

Numerical methods for contact and other geometrically non- linear problems Parallel solution of contact shape optimization problems with Coulomb friction based on domain decomposition P. Beremlijski, Tom´aˇsBrzobohat´y,Tom´aˇsKozubek and Alexandros Markopoulos ...... 28-2

Parallel solution of elasto-plastic problems Martin Cerm´akˇ and Michal Merta ...... 28-3

Convergence analysis for multilevel variance estimators in Multilevel Monte Carlo Methods and application for random obstacle problems Alexey Chernov and Claudio Bierig ...... 28-4

Scalable algorithms and conditioning of constraints arising from variationally consistent discretization of contact problems ZdenˇekDost´al, Tom´aˇsKozubek and Oldˇrich Vlach ...... 28-5

Parallel solution of contact problems based on TFETI ZdenˇekDost´al,Tom´aˇsBrzobohat´y,Tom´aˇsKozubek, Alexandros Markopoulos and Oldˇrich Vlach ...... 28-6

Local averaging of contact with non matching meshes Guillaume Drouet and Patrick Hild ...... 28-8

xxvii BE/FE Approximation of higher order for nonsmooth problems, effective quadrature, and time discretization by implicit Runge-Kutta methods Joachim Gwinner ...... 28-9

A discretization for dynamic large deformation contact problems of nonlinear hypere- lastic continua Ralf Kornhuber, Oliver Sander and Jonathan Youett ...... 28-10

Parallel Level Set Methods for Large Deformation Contact Problems Rolf Krause, Valentina Poletti, Roberto Croce and Petr Kotas ...... 28-11

Error estimators for a partially clamped plate problem with boundary elements Matthias Maischak ...... 28-12

Optimal active-set and spectral algorithms for the solution of 3D contact problems with anisotropic friction Luk´aˇsPosp´ıˇsil, ZdenˇekDost´aland Tom´aˇsKozubek ...... 28-13 hp-adaptive FEM with biorthogonal basis functions for elliptic obstacle problems Andreas Schr¨oder and Lothar Banz ...... 28-14

High order BEM for frictional contact problems Ernst P. Stephan ...... 28-15

Numerical methods for fully nonlinear elliptic equations Pseudo transient continuation and time marching methods for Monge-Amp`eretype Equations Gerard Awanou ...... 29-2

General full discretizations for center manifolds, here for fully nonlinear equations and FEMs Klaus B¨ohmer ...... 29-2

Numerical Solution of Monge-Amp`ereEquation on Domains Bounded by Piecewise Conics Oleg Davydov and Abid Saeed ...... 29-3

xxviii Discontinuous Galerkin Finite Element Differential Calculus and Applications Xiaobing Feng ...... 29-3

A Finite Element Method for Hamilton-Jacobi-Bellman equations Max Jensen and Iain Smears ...... 29-4

Adaptivity and fully nonlinear problems Omar Lakkis and Tristan Pryer ...... 29-4

Finite element methods for the Monge-Amp`ereequation Michael Neilan ...... 29-5

Numerical modeling of flow in subsurface reservoirs Multiphase flow through porous media: An adaptive control volume finite element method Peyman Mostaghimi, James R. Percival, Brendan S. Tollit, Stephen J. Neeth- ling, Gerard J. Gorman, Matthew D. Jackson, Christopher C. Pain and Jefferson L.M.A. Gomes ...... 30-2

Multipoint flux domain decomposition time-splitting methods on general grids Andr´esArrar´as, Laura Portero and Ivan Yotov ...... 30-3

An optimization approach to large scale simulations of fluid flows in fractured media with finite elements on nonconforming grids Stefano Berrone, Sandra Pieraccini and Stefano Scial`o ...... 30-4

Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations Thomas Carraro and Christian Goll ...... 30-5

Efficient Bayesian uncertainty quantification of subsurface flow models using nested sampling and sparse polynomial chaos surrogates Ahmed H. Elsheikh, Mary F. Wheeler and Ibrahim Hoteit ...... 30-6

High-order cut-cell techniques for numerical upscaling in porous media Christian Engwer ...... 30-7

xxix Adjoints of finite element models Patrick E. Farrell and Simon W. Funke ...... 30-8

A global Jacobian method for mortar discretizations of nonlinear porous media flows Benjamin Ganis, Mika Juntunen, Gergina Pencheva, Mary F. Wheeler and Ivan Yotov

...... 30-9

Direct numerical simulation of two-phase flow at the pore scale Ali Q Raeini, Branko Bijeljic and Martin J Blunt ...... 30-10

Modeling flow with nonplanar fractures Gurpreet Singh, Omar Al-Hinai, Gergina Pencheva and Mary F. Wheeler ...... 30-11

Computational Environments for Energy and Environmental Modeling in Porous Media Mary F. Wheeler ...... 30-11

Multiscale domain decomposition methods for porous media flow coupled with geome- chanics Ivan Yotov ...... 30-12

PDEs on Surfaces Unfitted finite element methods using bulk meshes for surface partial differential equa- tions Klaus Deckelnick, Charles M. Elliott and Tom Ranner ...... 31-2

Discontinuous Galerkin methods for surface PDEs Andreas Dedner, Pravin Madhavan and Bj¨ornStinner ...... 31-3

Pattern formation in morphogenesis on evolving biological surfaces: Theory, numerics and applications Anotida Madzvamuse, Raquel Barreira, Charles M. Elliott, Ammon J. Meir and Necibe Tuncer ...... 31-4

An ALE ESFEM for solving PDEs on evolving surfaces Vanessa Styles ...... 31-5

xxx Numerical Simulations of Chemotaxis-Driven PDEs on surfaces Stefan Turek and Andriy Sokolov ...... 31-6

Sensitivity analysis and optimization for fluid-structure inter- action problems Parameter Estimation in Fluid-Structure Interaction and Subsurface Flows Ahmed H. Elsheikh, Thomas Richter, Mary F. Wheeler and Thomas Wick ...... 32-2

Towards optimal control of large deformation FSI problems including contact and topol- ogy change Thomas Richter and Thomas Wick ...... 32-3

Calculation of sensitivities for fluid-structure interactions Thomas Wick and Winnifried Wollner ...... 32-3

Stochastic finite elements and PDEs A posteriori error estimation for stochastic Galerkin FEMs Alex Bespalov, Catherine Powell and David Silvester ...... 33-2

Weak truncation error estimates for elliptic PDEs with lognormal coefficients Julia Charrier and Arnaud Debussche ...... 33-3

Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar Conservation Laws Alexandre Ern, Olivier Le Maˆıtreand Julie Tryoen ...... 33-4

Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems Angela Kunoth ...... 33-5

Finite element approximation of the Cahn-Hilliard-Cook equation Stig Larsson ...... 33-5

Exploring Emerging Manycore Architectures for Uncertainty Quantification Through Embedded Stochastic Galerkin Methods Eric Phipps, H. Carter Edwards, Jonathan Hu and Jakob T. Ostien ...... 33-6

xxxi Adaptive, Sparse Quadratures for Bayesian Inverse Problems Claudia Schillings and Christoph Schwab ...... 33-7

Multilevel Markov chain Monte Carlo algorithms for uncertainty quantification in sub- surface flow Aretha Teckentrup ...... 33-7

Superconvergence in DG: analysis and recovery A New Lax-Wendroff Discontinuous Galerkin Method with Superconvergence Wei Guo, Jianxian Qiu and Jing-Mei Qiu ...... 34-2

Energy norm error estimation for averaged discontinuous Galerkin methods in one space dimension Ferenc Izs´ak ...... 34-2

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering: Practical Considera- tions When Applied to Visualization Robert M. Kirby ...... 34-3

Error Estimation for the Discontinous Galerkin Method Applied to Hyperbolic Con- servation Laws Lilia Krivodonova and Noel Chalmers ...... 34-4

Computationally Efficient Boundary Filtering Using Smoothness-Increasing Accuracy- Conserving (SIAC) Methods Xiaozhou Li ...... 34-5

Superconvergence of a HDG method for fractional diffusion problems Kassem Mustapha and Bernardo Cockburn ...... 34-6

Post-processing discontinuous Galerkin solutions to Volterra integro-differential equa- tions: Analysis and Simulations Jennifer K. Ryan and Kassem Mustapha ...... 34-7

xxxii Time-Domain Boundary Integral Equations A fully discrete Kirchhoff formula based on CQ and Galerkin BEM Lehel Banjai, Antonio Laliena and Francisco-Javier Sayas ...... 35-2

Time-domain FEM/BEM coupling Lehel Banjai, Volker Gruhne, Christian Lubich and Francisco-Javier Sayas ...... 35-3

Convolution-in-time approximations of TDBIEs Penny J Davies and Dugald B Duncan ...... 35-4

Quadrature Schemes and Adaptivity for 2D Time Domain Boundary Element Methods (TD-BEM) Matthias Gl¨afke and Matthias Maischak ...... 35-5

Solving the heat equation with a fast multipole Galerkin boundary element method Michael Messner, Johannes Tausch, Martin Schanz and Wolfgang Weiss ...... 35-6

A Generalized Convolution Quadrature with Variable Time Stepping Stefan A. Sauter ...... 35-7

Adaptive methods for retarded boundary integral equations Stefan A. Sauter and Alexander Veit ...... 35-7

BEM for Parabolic Phase Phange Problems with Moving Interfaces Johannes Tausch and Elizabeth Case ...... 35-8

A hybrid approach to the time marching solution of Maxwell’s equations Daniel S. Weile ...... 35-9

Using space-time Galerkin stability theory to define a robust collocation method for time-domain boundary integral equations in electromagnetics Elwin van ’t Wout, Duncan R. van der Heul, Harmen van der Ven and Kees Vuik ...... 35-10

xxxiii 1 Abstracts of talks of invited speakers

FINITE ELEMENT METHODS IN COASTAL OCEAN MODELING: SUCCESSES AND CHALLENGES Clint N Dawson

The University of Texas at Austin, Department of Aerospace Engineering and Engineering Mechanics, College of Engineering, 1 University Station C0600, Austin TX 78712, USA [email protected]

The coastal ocean contains a diversity of physical and biological processes, often oc- curring at vastly different scales. In this talk, we will outline some of these processes and their mathematical description. We will then discuss how finite element methods are used in coastal ocean modeling and recent research into improvements to these algorithms. We will also highlight some of the successes of these methods in simulat- ing complex events, such as hurricane storm surges. Finally, we will outline several interesting challenges which are ripe for future research.

FORTY YEARS OF THE CROUZEIX-RAVIART ELEMENT Susanne C. Brenner

Center for Computation and Technology and Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA [email protected]

The Crouzeix-Raviart element, which is the simplest example of a nonconforming finite element, was introduced in a 1973 paper on the Stokes equation. In this talk I will discuss old and new results related to the Crouzeix-Raviart element that illustrate the methodologies for handling various aspects of nonconforming finite element methods, such as a priori and a posteriori error analyses, and the design and analysis of fast solvers.

1-1 ADVANCES IN REDUCING THE MESH BURDEN IN COMPUTATIONAL MECHANICS APPLICATIONS TO FRACTURE AND SURGICAL SIMULATION St´ephaneP.A. Bordas a

Institute of Mechanics & Advanced Materials (IMAM), School of Engineering, Cardiff University, Queen’s Buildings, The Parade, CARDIFF CF24 3AA, Wales, UK. [email protected], [email protected]

This presentation will address recent advances in enriched numerical methods to sim- plify the treatment of evolving discontinuities in the field variables or their derivatives: cracks or material interfaces; and to treat geometrically intricate domains and their evolution The presentation will be composed of three parts:

1. advances in numerical methods aiming at simplifying the treatment of complex geometries; Two competing approaches coexist in the literature to simplify the solution of partial differential equations over domains of complex and/or evolving geome- tries. One focuses on streamlining the transition between computer aided design (CAD) data and the solution of problems over the corresponding domains. An example of this is isogeometric analysis [3] where the geometry description and the approximation of the field variables are tied, thus enabling an exact treatment of the boundary as well as simplifying eventual geometric design iterations. The second approach follows an orthogonal direction, where the geometry is un- coupled from the field variable discretisation, e.g. embedded boundary methods such as the structured extended finite element method of [2]. We will present results emanating from both lines of thought: isogeometric analy sis of three-dimensional structures [8] and embedded interfaces for complex ge- ometries including sharp edges and vertices [6]. Some of these methods were reviewed in [4] and a tutorial on isogeometric analysis and associated implemen- tation aspects (MATLAB) is provided in [7].

2. advances in enriched formulations for evolving discontinuities such as cracks The extended finite element method (XFEM) was introduced in [1] with, as a basis, the partition of unity enrichment method of [5]. The XFEM enables the simulation of evolving discontinuities without or with minimal remeshing. We present recent developments in the area and focus on tackling difficulties associated with the control of the conditioning number of the system, the control of the error in quantities of interest. We also discuss results on the smoothed (extended) finite element method.

3. applications of those methods to problems in multi-crack growth and brain surgery simulation.

1-2 Finally, as an application, we will present a simple method to grow several hun- dreds of cracks in two-dimensions in order to predict their growth and coalescence in brittle materials using the XFEM. We will also present recent results permitting the simulation of cutting and con- tact during brain surgery simulation at 30 frames per second using an implicit time integration method and a hybrid, asynchronous CPU/GPU solver. We finish the presentation by conclusions and propositions for future work.

Acknowledgements St´ephaneP.A. Bordas wishes to thank the organisers of MAFELAP for their kind in- vitation. He is grateful to the European Research Council for funding the research presented (ERC Stg grant agreement No. 279578: “RealTCut Towards real time mul- tiscale simulation of cutting in non-linear materials with applications to surgical sim- ulation and computer guided surgery”).

References

[1] Ted Belytschko and T Black. Elastic crack growth in finite elements with minimal remeshing. International journal for numerical methods in engineering, 45(5):601– 620, 1999. [2] Ted Belytschko, Chandu Parimi, Nicolas Mo¨es,N Sukumar, and Shuji Usui. Struc- tured extended finite element methods for solids defined by implicit surfaces. In- ternational journal for numerical methods in engineering, 56(4):609–635, 2003. [3] Thomas JR Hughes, John A Cottrell, and Yuri Bazilevs. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement. Computer meth- ods in applied mechanics and engineering, 194(39):4135–4195, 2005. [4] H Lian, SPA Bordas, R Sevilla, and RN Simpson. Recent developments in cad/analysis integration. arXiv preprint arXiv:1210.8216, 2012. [5] Jens Markus Melenk and Ivo Babuˇska. The partition of unity finite element method: basic theory and applications. Computer methods in applied mechanics and engi- neering, 139(1):289–314, 1996. [6] Mohammed Moumnassi, Salim Belouettar, Eric´ B´echet, St´ephaneBordas, Didier Quoirin, and Michel Potier-Ferry. Finite element analysis on implicitly defined do- mains: An accurate representation based on arbitrary parametric surfaces. Com- puter methods in applied mechanics and engineering, 200(5):774–796, 2011. [7] Vinh Phu Nguyen, Robert N Simpson, St´ephaneBordas, and Timon Rabczuk. An introduction to isogeometric analysis with matlab\ textsu perscript {\ tex- tregistered {}} implementation: Fem and xfem formulations. arXiv preprint arXiv:1205.2129, 2012. [8] MA Scott, RN Simpson, JA Evans, S Lipton, SPA Bordas, TJR Hughes, and TW Sederberg. Isogeometric boundary element analysis using unstructured t- splines. Computer Methods in Applied Mechanics and Engineering, 254:197221, 2012.

1-3 FIELDS, CONTROL FIELDS, AND COMMUTING DIAGRAMS IN ISOGEOMETRIC ANALYSIS Annalisa Buffa1, Giancarlo Sangalli2 and Rafael V´azquez1

1 IMATI CNR “E. Magenes”, Via Ferrata 1, 27100 Pavia, Italy [email protected] 2 Universit`adegli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy

The numerical discretization of equations enjoying a relevant geometric structure, is one of the most interesting challenge of numerical analysis for PDEs and several results have been obtained in the last decade. On the one hand, discrete schemes have to preserve the geometric structure of the underlying PDEs in order to avoid spurious behaviors, instability or non-physical solutions: e.g., in electromagnetics, numerical schemes has to be related with a discrete De Rham complex. On the other hand, especially in view of high frequency computations, numerical schemes need to be efficient and accurate. In this talk, I will present splines approximations of the vector fields and I will discuss the properties of the spline discretization of the De Rham complex. I will show the relation between the spline complex and the topology of the knot mesh and of the control net. I will address the construction of canonical basis for spline spaces of vector fields, and possibly discuss the extension to NURBS. I will finish my talk with a few numerical results.

References

[1] A. Buffa, G. Sangalli, R. Vazquez´ , Isogeometric Methods for Computa- tional Electromagnetics: B-spline and T-spline discretizations, J. Comput. Phys., submitted.

[2] A. Buffa, J. Rivas, G. Sangalli, R. Vazquez´ , Isogeometric Discrete Differ- ential Forms in Three Dimensions, SIAM J. Numer. Anal. 49 (2011), pp. 818-842.

[3] A. Buffa, G. Sangalli, R. Vazquez´ , Isogeometric analysis in electromagnet- ics: B-splines approximation , Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 17-20, pp. 11431152.

1-4 THE INF-SUP CONSTANT OF THE DIVERGENCE Martin Costabel

IRMAR, Universit´ede Rennes 1, Rennes, France [email protected]

The inf-sup condition for the divergence, under different disguises variously attributed to Lions, Neˇcas,Babuˇska-Aziz or Ladyzhenskaya, Babuˇska and Brezzi, states the posi- tivity of the inf-sup constant of a domain Ω, or equivalently, the existence of a bounded 1 right inverse of the divergence operator mapping the Sobolev space H0 (Ω) to the space of functions square integrable on Ω with mean value zero. The condition has been known to hold for bounded Lipschitz domains for half a century, and its close relation to Korn’s second inequality, to the spectrum of the Schur complement of the Stokes system and to the century-old Cosserat eigenvalue problem has been known for a very long time, too. Yet, despite this venerable history and despite its permanent place in the spotlight because of its importance for stability and efficiency estimates in fluid dynamics, the dependence of the constant on the domain is a subject where still many basic questions remain open. Others have recently found answers or have seen progress towards answers. The talk will trace some of this progress and present results obtained together with Monique Dauge and in collaboration with Christine Bernardi and Vivette Girault.

1-5 FINITE ELEMENT METHODS FOR SURFACE PDES Charles M. Elliott

Mathematics Institute and Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, UK [email protected]

Surface partial differential equations arise in a wide variety of applications:- surfactants in two phase flow, pattern formation on growing biological surfaces, surface phase sep- aration, diffusion along fractures in porous media, modelling of biomembranes etc. They are examples of partial differential equations on manifolds. As such they provide a remaining challenge within the general subject of the numerical analysis of partial differential equations. The framework is essentially geometric because the domain in which the equation holds is curved. They are linked naturally to the geometric equa- tions for surfaces such as the minimal surface equation, motion by mean curvature and Willmore flow. In this talk we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we need geometric analysis and, in the context of evolving surfaces, transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis will be presented along with illustrative numeri- cal examples. The topic is also the subject of a mini-symposium at this conference. I will report on work primarily with Gerd Dziuk. Other collaborators are Klaus Deck- elnick, Tom Ranner, Bjorn Stinner, Chandrashekar Venkataramaran, Vanessa Styles and Anotida Madzvamuse.

1-6 A STOCHASTIC COLLOCATION APPROACH TO PDE-CONSTRAINED OPTIMIZATION FOR RANDOM DATA IDENTIFICATION PROBLEMS Max Gunzburger

Department of Scientific Computing, Florida State University, Tallahassee FL 32306-4120, USA [email protected]

We present a scalable, embarrassingly parallel mechanism for optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole prob- ability distribution of input random data, given the probability distribution of some response (quantity of physical interest) of a system of partial differential equations (PDEs). The stochastic inverse problem can be described by an objective functional constrained by a system of stochastic PDEs. Several identification objectives are dis- cussed that either minimize the expectation of a tracking cost functional or minimize the difference of desired statistical quantities in the appropriate norms. The distributed parameters/controls can be either deterministic or stochastic. Given an objective, we prove the existence of an optimal solution, establish the validity of the Lagrange multi- plier rule, and obtain a stochastic optimality system of equations. To characterize data with moderately large amounts of uncertainty, we introduce a novel stochastic param- eter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation finite element method. This allows for de- coupled, moderately high-dimensional, parameterized computations of the stochastic optimality system, where at each collocation point deterministic analyses and tech- niques can be utilized. The rigorously derived error estimates for the fully discrete problems are described and used to compare the efficiency of the method with that of several other techniques. Numerical examples illustrate the theoretical results and demonstrate the distinctions between the various stochastic identification objectives. This is joint work with Catalin Trenchea (University of Pittsburgh) and Clayton Webster (Oak Ridge National Laboratory).

1-7 TIME DOMAIN INTEGRAL EQUATIONS FOR COMPUTATIONAL ELECTROMAGNETISM Peter Monk

Mathematical Sciences, University of Delaware, Newark DE 19716, USA [email protected]

Scattering problems for Maxwell’s equations can be solved in the frequency or time domain. In the frequency domain both finite element and boundary integral methods are in common use, and their relative strengths and weaknesses are well understood. In contrast, in the time domain the principal technique is the finite difference time domain method (or the Discontinuous Galerkin Method). However, time domain in- tegral equations have become much more popular in recent years, although they still represent a considerable coding challenge. This can be mitigated by using the convo- lution quadrature approach (CQ) [see C. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Nu- mer. Math., 67 (1994), pp. 365389.], together with a boundary Galerkin method in space and efficient integral equation software such as BEM++ [see W. Smigaj,´ S. Arridge, T. Betcke, J. Phillips, M. Schweiger, “Solving Boundary Integral Problems with BEM++”, ACM Trans. Math. Software]. The result is a convenient and robust method. I shall outline the CQ method applied to Maxwell’s equations using the problem of computing waves scattered by a penetrable object as a model problem. After discussing some theoretical aspects such as error estimates and other properties of the scheme, I shall present numerical results computed using the recently developed time domain toolbox within the BEM++ library.

1-8 THE EMERGENCE OF PREDICTIVE COMPUTATIONAL SCIENCE: VALIDATION AND VERIFICATION OF COMPUTATIONAL MODELS OF COMPLEX PHYSICAL SYSTEMS J. Tinsley Oden

Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin, Texas, USA [email protected]

Predictive Science is understood to be the scientific discipline concerned with the use of mathematical and computational models to forecast physical events. It embraces the processes of model selection, calibration, validation, verification, and their use in forecasting features of physical events with quantified uncertainty. It is argued that the principles of predictive science and the tools it employs are rooted in the philosophical foundations of science itself and that they evoke a need for reviewing exactly how scientific knowledge is obtained and how it is interpreted in a statistical setting. We adopt a Bayesian framework to discuss these issues and to describe methods of statistical calibration, model plausibility, validation, and solution of stochastic systems. We explore several areas fundamental to contemporary compu- tational science: coarse-gaining and validation of molecular models, maximum entropy methods, experimental design based on Shannon information theoretics, virtual valida- tion of models, model bias or inadequacy, and model selection. Finite element methods enter in the analysis of macroscale models of validation experiments. Examples from nanomanufacturing are discussed.

1-9 WHAT IS THE LARGEST FINITE ELEMENT SYSTEM THAT CAN BE SOLVED TODAY? Ulrich Ruede

University Erlangen-Nuremberg, Department of Computer Science 10, Cauerstraße 11, D-91058 Erlangen, Germany [email protected]

Top supercomputers have progressed beyond the Peta-Scale, i.e. they are capable to perform in excess of 1015 operations per second and their main memory approaches that scale. However, this performance can only be achieved with massively parallel systems that have several hundred thousand processor cores, posing severe challenges to algorithm and software development. The fastest FE solvers, such as multigrid methods, scale linearly in numerical complexity with modest constants, but there has been a debate whether they do not suffer from sequential bottlenecks. In this talk we will present our experience in implementing multigrid solvers for FE problems with up to a trillion (1012) unknowns. This is e.g. enough to discretize the whole volume of planet for simulating the the Earth mantle convection problem with a global resolution of about 1km. The compute times are then around 1 minute for computing a single flow field, so that the algorithm can still be used reasonably within an implicit time stepping procedure.

DOUBLE COMPLEXES AND LOCAL BOUNDED COCHAIN PROJECTIONS Ragnar Winther

Centre of Mathematics for Applications (CMA), University of Oslo. CMA c/o Dept of Mathematics, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway [email protected]

The construction of bounded projections is a key tool for establishing stability of various finite element methods. In this talk we discuss a new construction of projections from HΛk, the space of differential k forms which belong to L2 and whose exterior derivative also belongs to L2, to finite dimensional subspaces consisting of piecewise polynomial differential forms defined on a simplicial mesh of the domain. Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom. Moreover, these projections commute with the exterior derivative and are bounded in the HΛk norm independent of the mesh size. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Cl´ement interpolant. A double complex structure is introduced as a key tool to carry out the construction.

1-10 2: Parallel session talks

2 Abstracts of talks in parallel sessions

2-1 2: Parallel session talks

VERIFICATION OF FUNCTIONAL A POSTERIORI ERROR ESTIMATES FOR OBSTACLE PROBLEM Petr Harasim and Jan Valdmana

Centre of Excellence IT4Innovations, VSB-Technicalˇ University of Ostrava, Czech Republic [email protected],

We verify functional a posteriori error estimates proposed by S. Repin for the case of an obstacle problem. Quality of a numerical solution obtained by the finite element method is compared with the exact solution (obtained in 1D case) to demonstrate the sharpness of the studied error estimated. Extention to 2D case will be also reported.

2-2 2: Parallel session talks

CONVERGENCE OF HP -FEM IN THREE DIMENSIONAL COMPUTATION OF THERMOELECTRIC EFFECTS Razi Abdul-Rahmana and Sarah Kamaludinb

School of Mechanical Engineering, Universiti Sains Malaysia, Penang, Malaysia. [email protected], [email protected]

Solutions to coupled-field problems like thermoelecric effects are traditionally computed with a partitioned-based approach where data exchanges between respective domains must be carefully treated to ensure convergence of solutions. Alternatively, a monolithic framework to describe the coupled fields is arguable better amenable to adaptive finite element solutions. A difficulty arises for many cases where the resulting system matrices are nonsymmetrical and negative definite. Resorting to nonlinear solvers is costly for adaptive solutions. Nevertheless, under certain assumptions of the thermoelectric effects, it is possible to model them as a linear, symmetric system of coupled PDEs. We present such an approach following the framework which is based on the Onsagers reciprocity theorem elaborated in [C. Vokas and M. Kasper. Adaptation in multinature problems. In: Proceedings of the 13th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Graz, 2008]. In our implementation, only the constitutive relations involving the Peltier and Seebeck effects are considered to relate heat and electric potential. By assuming negligible nonlinear effects of Thomson effect and Joule heating, a linearization of the constitutive thermoelectric equations with respect to a reference temperature is possible with application of the Onsagers reciprocity theorem. Computational performance and convergence of hp-adaptive solutions in three di- mensional problems based on the framework is of interest. The hp-adaptivity is imple- mented to simultaneously reduce the discretization errors of the two physical natures, i.e., temperature and electric potential, in a similar manner as in a single nature prob- lem. An a posteriori error estimator based on element-wise residuals and jumps at element boundaries is used. In the current implementation, h or p-extension is trig- gered if any of the two natures has not reached a prescribed error limit. Hence, due to the direct formulation of the coupled problem, the local solution of one nature may be over-evaluated relative to the other one. Nevertheless, the convergence of errors in energy norm for both natures is found to be reasonble. Our implementation strongly suggests that the thermoelectric problems in three dimensions may well be computed with the hp-FEM so that accurate results can be achieved without an excessive use of computational resources when treating them as nonlinear problems, for example as in [J. L. P´erez-Aparicio,R. L. Taylor and D. Gavela. Finite element analysis of nonlinear fully coupled thermoelectric materials. Computational Mechanics, 40: 35-45, 2007]. Thus, it allows fast hp-FEM computations while preserving accuracy within a range of practical problems.

2-3 2: Parallel session talks

DISCONTINUOUS GALERKIN TIME STEPPING SCHEMES COMBINED WITH LOCAL PROJECTION STABILIZATION METHODS APPLIED TO TRANSIENT STOKES PROBLEMS: STABILITY AND CONVERGENCE Naveed Ahmeda and Gunar Matthies

Universit¨atKassel, Fachbereich 10 Mathematik und Naturwissenschaften, Institut f¨urMathematik, Heinrich-Plett-Straße 40, 34132 Kassel, Germany [email protected]

We will consider finite element methods for solving transient Stokes problems in the case of equal order interpolation of velocity and pressure. Since these pairs do not satisfy an inf-sup condition, a spatial stabilization of the pressure is needed. We will apply a stabilization term based on the one-level version of local projection method. Since projection space and ansatz space are defined on the same mesh, no coupling of degrees of freedom not belonging to the same mesh cell is introduced. This is in contrast to the continuous interior penalty method, the subgrid scale modeling, the orthogonal subgrid method and the two-level local projection stabilization method. Our main interest lies in the combination of local projection methods in space and discontinuous Galerkin methods of degree k in time. We will derive the unconditional stability of the method and give error estimates for the semi discrete and for the fully discrete problem. Our numerical examples will show that dG(k) methods are accurate of order k + 1 in the L2(L2)-norm for velocity and pressure. The dG-norm which consists of the integrated LPS-norm and the jumps at the discrete time points shows a convergence of order k + 1/2 in time. At the discrete time points, the error in the velocity is superconvergent of order 2k + 1 for dG(k). By means of a simple post-processing, the solutions of dG(k) can be improved such that the convergence order k + 2 is obtained for the L2(L2)-norm of the velocity. At the discrete time points, the post-processing provided for dG(k) a superconvergence of order 2k + 1 also for the pressure. The choice of stabilization parameters plays a critical role in the success of stabilized methods. Our numerical studies will suggest how to choose stabilization parameters. We also verify for higher order time discretization that, for small time steps, the pressure is stable for initial data that are not discretely divergence free. The research was supported by DFG through grant MA 4713/2 − 1.

2-4 2: Parallel session talks

ANISOTROPIC MESH ADAPTATION FOR THE AMBROSIO-TORTORELLI MODEL: APPLICATION TO QUASI-STATIC CRACK PROPAGATION Marco Artina1a, Massimo Fonrasier1b, Simona Perotto2c and Stefano Micheletti2d

1Faculty of Mathematics, Technische Universit¨atM¨unchen, Boltzmannstrasse 3, 85748, Garching, Germany [email protected], [email protected], 2MOX - Modeling and Scientific Computing, Dipartimento di Matematica ”F.Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy. [email protected], [email protected]

The minimization of the Mumford-Shah functional represents a very challenging issue since it is non-smooth and non-convex. This functional characterizes several problems. In particular, we are interested in the Francfort-Marigo model in the context of the quasi-static fracture propagation.

To numerically approximate the problem, one needs first to Γ-approximate the non-smooth energy, which depends on the displacement and on its discontinuity set, by using a smoother version (i.e. the model proposed by Ambrosio and Tortorelli) where a smooth indicator function identifies the discontinuity set. Then, we resort to an adaptive finite element approach based on piecewise linear elements. Nevertheless, similarly to early work by Chambolle et al. but differently from recent approaches by S¨uliet al. where isotropic meshes are used, in this work we investigate how anisotropic meshes can lead to significant improvements in terms of the balance between accuracy and complexity. Indeed, the main advantage achievable is a significant reduction of the number of elements to capture with good confidence the expected fracture path. The employment of anisotropic grids allows us to shortly follow the propagation of the fracture by refining it only in a very thin neighborhood of the crack.

In this talk, we first present the derivation of a novel anisotropic a posteriori error estimator driving the mesh adaptation for the approximation of the Ambrosio-Tortorelli model. Then, we provide several numerical results which corroborate the accuracy as well as the computational saving led by an anisotropic mesh adaptation procedure.

2-5 2: Parallel session talks

A POSTERIORI ERROR ANALYSIS FOR TIME-DEPENDENT STOKES EQUATIONS Eberhard B¨ansch1, Johan Jansson2, Fotini Karakatsani3 and Charalambos Makridakis4

1Chair of Applied Mathematics III, University of Erlangen-Nuremberg, Cauerstr. 11 91058 Erlangen, Germany, [email protected] 2BCAM - Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao Basque Country, Spain, [email protected] 3Department of Mathematics and Statistics, University of Strathclyde, 16 Richmond Street, Glasgow G1 1XQ, United Kingdom, [email protected] 4Department of Applied Mathematics, University of Crete, L. Knosou GR 71409 Heraklion, Greece, [email protected]

We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations of time-dependent Stokes problem. The time discretization uses the backward Euler method and the spatial discretization uses finite element spaces that are allowed to change in time. The a posteriori error estimates are derived by applying the reconstruction technique.

References

[1] E. B¨ansch, J. Jansson, F. Karakatsani, Ch. Makridakis, On the a posteriori error control of time dependent Stokes equations, preprint (2013).

[2] F. Karakatsani, Ch. Makridakis, A posteriori estimates for approximations of time dependent Stokes equations, IMA J. Numer. Anal. 27 (2007) 741–764.

[3] O. Lakkis, Ch. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp., 75 (2006) 1627-1658.

2-6 2: Parallel session talks

NON-CONFORMING FINITE ELEMENT METHODS FOR THE OBSTACLE PROBLEM Carsten Carstensen and Karoline K¨ohler

Institut f¨urMathematik, Humboldt-Universit¨atzu Berlin, Unter den Linden 6, D-10099 Berlin, Germany [email protected]

In this talk we will present a priori and a posteriori error analysis for the non-conforming Crouzeix-Raviart finite element method (FEM). We consider general obstacles χ ∈ H2(Ω) on bounded polygonal domains Ω ⊆ R2 and non-homogeneous Dirichlet bound- ary conditions. Under standard regularity assumptions u ∈ H2(Ω) the Crouzeix- Raviart finite element solution allows for a best-approximation result for the gradients plus additional terms which converge linearly as the maximal mesh size approaches zero. Reliable and efficient control over the error follows from residual based a poste- riori error analysis. The design of a discrete Lagrange multiplier leads to a guaranteed upper bound for the exact error. The Crouzeix-Raviart FEM allows for the computa- tion of a guaranteed lower bound for the exact minimal energy, which in turn can be employed to present an alternative a posteriori error estimate. Numerical experiments investigate the practical performance of a related adaptive algorithm and explore the accuracy of upper and lower energy bounds.

2-7 2: Parallel session talks

ON THE NUMERICAL SIMULATION OF WELL STABILITY Philippe R B Devloo1, Erick Raggio Slis Santos2 and Diogo Lira Cec´ılio3

1 Departamento de Estruturas - Faculdade de Engenharia Civil - Unicamp, Brazil [email protected] 2 CENPES - Petrobras, Brazil 3 Departamento de Engenharia de Petr´oleo- Faculdade de Engenharia Mecˆanica- Unicamp, Brazil

A finite element simulation is presented for analysing the stability of excavated wells through the use of the Sandler DiMaggio elastoplastic model. In a first step the geolog- ical rock is stressed to the effective in-situ stress state. The stress state of the excavated well is obtained by a gradual stress state transfer at the boundary of the well from the original. The resulting stress state shows plastic behaviour in the region around the wellbore. The finite element mesh is hp-refined around the wellbore to enhance the precision of the elastoplastic simulation. The second invariant of the plastic deformation tensor is USED as damage criterion to decide on material collapse. Material collapse is represented numerically by the removal of elements. In order to apply adaptive methods to elastoplastic simulations, general purpose procedures were developed to transfer the elastoplastic deformation history from one mesh to another. The Sandler DiMaggio elastoplastic material model was implemented in a general purpose object oriented framework which computes the elastoplastic response of a material point based on the definition of the yield surface and hardening law. The tangent matrix of the stress strain relation is computed consistently by the use of automatic differentiation. Unstable wells show increase of plastic deformation as the material is removed. Stable wells reduce localized plastic deformation as the well geometry is modified. This model represents advances in terms of well stability analyses in the excavation and production phases.

2-8 2: Parallel session talks

ESTIMATION OF DISCRETIZATION AND ALGEBRAIC ERROR VIA QUASI-EQUILIBRATED FLUXES FOR DISCONTINUOUS GALERKIN METHODS V´ıtDolejˇs´ı1a, Ivana Sebestov´aˇ 1b and Martin Vohral´ık2

1Charles University in Prague, Faculty of Mathematics and Physics, Sokolovsk´a83, 186 75 Praha 8, Czech Republic. [email protected], [email protected] 2INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France. [email protected]

We present the generalization of guaranteed and locally efficient a posteriori error estimates based on quasi-equilibration of fluxes reconstruction for interior penalty dis- continuous Galerkin methods on simplicial meshes. The estimation newly involves the algebraic error arising from the inexactness in solving underlying algebraic problem. It is measured via an algebraic error flux reconstruction. Therefore, the flux reconstruc- tion as the total error consists of individual components: discretization and algebraic error reconstructions. We focus on exploiting the main advantage of DGM, namely decoupling of the global problem into element-wise problems. Firstly, variable polynomial degree of the approx- imate solution is permitted. Secondly, hanging nodes are allowed in our setting and the flux reconstruction is constructed in broken Raviart–Thomas–N´ed´elec space avoiding enforcing continuity of normal traces as well as constructing of matching submeshes. Moreover, following an approach proposed in [1], the algebraic component is treated by performing some additional steps of the iterative algebraic solver and subsequently by comparing the discretization error at two iteration steps of linear algebra computation. Such a construction is inexpensive however only results in quasi-equilibration. Further, we point out that the factual construction of considered reconstructions is not needed and as such evaluation of estimators is not costly. Using derived estimates, we propose stopping criteria for iterative algebraic solvers and subsequently an adaptive strategy for solving a linear elliptic equation. Finally, we present numerical results demonstrating good prediction of distribution of both discretization and algebraic components and significant computational savings in com- parison with classical approaches.

References

[1] V. Dolejˇs´ı,I. Sebestov´a,andˇ M. Vohral´ık. Discretization and algebraic error esti- mation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids. In preparation.

2-9 2: Parallel session talks

CONDENSING THE SPECTRAL ELEMENT METHOD FOR TIME DOMAIN WAVE PROBLEMS Dugald B Duncana and Mark Payneb

Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, UK. [email protected], [email protected]

The Spectral Element Method (SEM) is widely used in the approximation of time do- main linear wave propagation problems posed in second order form, e.g. the acoustic and elastic waves equations. At its heart is a clever combination of node-based poly- nomial approximation within each element with a quadrature rule using the internal node locations as quadrature points to give a diagonal mass matrix, hence making the scheme explicit. The clever part is in choosing the node locations to give high order accuracy and good stability properties. In 1d, the SEM consists of one set of explicit difference equations on three time levels at “main nodes” located at the ends of the elements and linked to points in the elements around them, and other sets of equations at “internal nodes” which are not so widely linked, sometimes only to points within their own element. In 2d, nodes on element edges have internal element linkages in one direction and wider linkages in the other and this generalises further in 3d. All of this is explained comprehensively in Cohen’s book “Higher-order numerical methods for transient wave equations” (Springer 2002). In this talk we present a different view of the SEM obtained after condensing out the internal nodes to leave a multilevel scheme on the main nodes only. We show the result, perhaps surprising, that the relatively popular SEMs are actually equivalent to little-used (indeed deprecated) multilevel “symmetric schemes” and so they have the same properties. We show that recasting the SEM in this form can make some of the analysis easier, we compare the efficiency of implementing the two approaches and derive improved SEM-type schemes using the condensed form to guide the process.

2-10 2: Parallel session talks

LOCAL MASS CONSERVATION OF STOKES FINITE ELEMENTS Daniele Boffi1a, Nicola Cavallini1b Francesca Gardini1c and Lucia Gastaldi2

1Dipartimento di Matematica, Univerist`adi Pavia, Pavia, Italy. [email protected], [email protected] [email protected] 2Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Universit`adi Brescia, Brescia, Italy. [email protected]

It is well-known that in the approximation of the solution of the Stokes problem the incompressibility constraint is imposed in a weak sense, so that the divergence free condition is not imposed exactly at the discrete level, unless the divergence of the velocity space is contained in the pressure space. In this talk we discuss the stability of some Stokes finite elements, which allow for a more local enforcement of the divergence free condition. In particular, we consider a modification of Hood-Taylor and Bercovier-Pironneau schemes which consists in adding piecewise constant functions to the pressure space. This enhancement, which had been already used in the literature, is driven by the goal of achieving an improved mass conservation at element level. We prove the inf-sup condition for the enhanced spaces in a general setting and we present some numerical tests which confirm the stability properties and the improvement in the local mass conservation (see [2]). Moreover, we analyze the pressure enhancement modification in the framework of lowest order stabilized finite elements as well (see [1]). The main consequence of the analysis is that the enhancement is poorly effective in the case of low order elements and non smooth data. In particular, if the polynomial order of the velocity space is not high enough (quadratic in 2D or cubic in 3D), then the pressure enhancement requires an appropriate stabilization involving the pressure jumps along the interelements. First of all, we observe that such stabilization destroys the local nature of the mass conservation property. Moreover, the stabilization term introduces a consistency error with reduced rate of convergence in case of non-smooth pressure solutions. This drawback applies, for instance, to the stabilized P1 −P0 element or to the enhanced version of the popular P1 − P1 stabilized element. In order to circumvent this phenomena we introduce a new finite element that combines the feasible characteristics of the P1−P1 stabilized element and mass conservation properties of the Bercovier-Pironneau element. One of the motivations for this research is the application of the proposed schemes to the finite element Immersed Boundary Method for the approximation of fluid-structure interaction problems (see [3]).

References

[1] D. Boffi, N. Cavallini, F. Gardini, L. Gastaldi: “Stabilized stokes element and local mass conservation”, Bollettino U.M.I., (9) V (2012), no. 3, 543-573.

2-11 2: Parallel session talks

[2] D. Boffi, N. Cavallini, F. Gardini, L. Gastaldi: “Local mass conservation of Stokes finite elements”, J. Sci. Comput., 52 (2012), no. 2, 383-400.

[3] D. Boffi, N. Cavallini, F. Gardini, L. Gastaldi: “Immersed boundary method: per- formance analysis of popular finite element spaces”, in:Coupled Problems 2011, pp. 1-12, M. Papadrakakis, E. Onate and B. Schrefler (Eds). Cimne.

ENRICHING A HANKEL BASIS BY RAY TRACING IN THE ULTRA WEAK VARIATIONAL FORMULATION C. J. Howarth1a, Simon Chandler-Wilde1, Stephen Langdon1 and P.N. Childs.2

1University of Reading, Reading, UK. [email protected] 2Schlumberger Gould Research, Cambridge, UK.

The Ultra Weak Variational Formulation (UWVF) is a new generation finite element method for approximating time harmonic acoustic and electromagnetic wave propaga- tion. The UWVF assumes wave like behaviour on each element, but otherwise allows flexibility in the approximation space. We exploit this flexibility by combining the numerical method with ray tracing solutions, in order to find accurate solutions at a lower computational cost than the standard UWVF. Time harmonic acoustic wave scattering is modelled in 2D by the Helmholtz prob- lem where Ω is a polygonal domain with boundary Γ. The UWVF approximation takes the form of a linear combination of basis functions upon each element Ωk. Basis func- tions are required to solve the homogeneous Helmholtz equation, so incorporating the oscillatory behaviour of the solution. Much current literature uses an equally spaced plane wave basis on each element. As an alternative, we instead use a Hankel basis: cylindrical waves originating from source points yk,l ∈/ Ωk. The choice of yk,l allows flexibility in both the direction and the level of curvature of the basis function over the element. At high frequencies a ray model gives a good understanding of the direction of propagation of a wave. We use the ideas of ray tracing to find a good a-priori choice of basis function. Consider a domain Ω enclosing a smooth, convex scatterer with no straight edges, the boundary of which we denote by Γ. Let the incident field ui be a plane wave and the wavenumber κ be constant in Ω: ray directions perpendicular to the wavefronts are parallel, until they reach the scatterer, reflect, and continue in a straight line. For any given point in the illuminated region x ∈ Ω, by tracing the ray through this point back, we can find the point of interaction z ∈ Γ where the incident ray hits the scatterer, and the angle of reflection θr. For points inside an element Ωk, we consider the local scattered field to be originat- ing from a single centre of curvature, which we can find by considering the intersection of rays from points which are close to one another, x and x0. If we extend the rays that travel through x and x0 back through the scatterer, they will cross at some point P either within or on the opposite side of the scatterer. Taking the limit as x → x0, we take this intersection point as the centre of curvature xC.

2-12 2: Parallel session talks

The UWVF can be extended to incorporate the ray traced directions and centres of curvature into the Hankel basis, using just two basis functions per element: one a single plane wave representing the incident field and the other a point source centred at xC, found by the ray tracing algorithm representing the scattered field. Using just the two ray traced basis functions per element for a circular scatterer, for κ = 80, we get an L2 relative error of under 9% using 0.4 degrees of freedom per wavelength. If only a more general idea of the wave interaction is needed rather than high accuracy, perhaps as an initial guess of state, then this approach suggests potential computational savings compared to more standard methods. To achieve higher accuracy we require more basis functions per element. In addition to including the ray traced basis representing the incident field direction, we also include further directions, equally spaced around a circle of radius R  1 (thus simulating plane waves), together with a final point source centered at the center of curvature xC given by our ray tracing algorithm. Including the ray traced basis functions leads to a reduction in the overall number of degrees of freedom required to achieve a given level of accuracy.

2-13 2: Parallel session talks

ERROR ESTIMATES FOR NONLINEAR CONVECTIVE AND SINGULARLY PERTURBED PROBLEMS IN FINITE ELEMENT METHODS V´aclav Kuˇcera

Department of Numerical Mathematics, Charles University in Prague, Sokolovsk´a83, Praha 8, 186 75, Czech republic. [email protected]

We are concerned with the analysis of the discontinuous Galerkin (DG) and standard conforming finite element methods applied to the nonstationary convective or singularly perturbed convection-diffusion problem defined in Ω ⊂ Rd: ∂u a) + div f(u) = ε∆u + g in Ω × (0,T ), ∂t ∂u b) u = uD, ε = gN , ΓD×(0,T ) ∂n ΓN ×(0,T ) c) u(x, 0) = u0(x), x ∈ Ω.

We derive apriori error estimates in the L∞(L2)-norm which are uniform with re- spect to ε → 0 and are valid even for the limiting case ε = 0. For various explicit schemes, such an error analysis was presented in a series of papers starting with [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontin- uous Galerkin methods for scalar conservation laws, 2004]. There the argument relies on a nonstandard estimate of the convective terms using E-fluxes and a mathematical induction argument with respect to time. Thus the technique cannot be directly ap- plied to estimates for the method of lines (no discrete structure with respect to time) and implicit discretizations (not enough apriori information about the solution on the next time level). In the present work, we circumvent these obstacles: Method of lines. Here we apply two different techniques. First, we use the so-called continuous mathematical induction [Y. R. Chao, A note on Continuous mathemati- cal induction, 1919] instead of standard mathematical induction used for the explicit schemes. This is a technique that we shall also use in the implicit case. Alternatively, we prove the same result using a nonlinear variant of Gronwall’s inequality. We prove that the latter technique has no discrete counterpart for implicit time discretizations. Implicit time discretization. First, we prove that for the implicit scheme, the de- sired error estimates cannot be proved only from the error equation and the considered estimates of its individual terms. To obtain more information about the problem, we 2 n introduce a continuation eh : [0,T ] → L (Ω) of the error eh, n = 0, ··· ,N constructed by means of a suitable modification of the discrete problem. We prove that estimates for this continuated solution directly imply estimates for the original implicit solution. The fact that eh is continuous with respect to time and that it relates to the struc- ture of the problem allows us to prove estimates for eh via continuous mathematical induction. A principal artefact of the technique is that we obtain a rather restrictive CFL-like condition even in the case of an implicit time discretization and that the result is not

2-14 2: Parallel session talks valid for the lowest order approximation degrees (we need p > 1 + d/2). The technique can be straightforwardly generalized to much more complicated problems.

A DOMAIN DECOMPOSITION METHOD WITH AN OPTIMIZED PENALTY PARAMETER Chang-Ock Lee1 and Eun-Hee Park2

1 Department of Mathematical Sciences, KAIST, Daejeon, South Korea [email protected] 2 Division of Computational Mathematics, National Institute for Mathematical Sciences, Daejeon, South Korea [email protected]

The dual-primal finite element tearing and interconnecting (FETI-DP) method [1] is a domain decomposition method, which enforces the continuity across the subdomain in- terface by Lagrange multipliers. A dual iterative substructuring method with a penalty term was introduced in the previous works by the authors [2, 3], which is a variant of the FETI-DP method in terms of the way to deal with the continuity on the interface. The proposed method imposes the continuity not only by using Lagrange multipliers but also by adding a penalty term which consists of a positive penalty parameter η and a measure of the jump across the interface. Due to the penalty term, the proposed it- erative method has a better convergence property than the standard FETI-DP method in the sense that the condition number of the resultant dual problem is bounded by a constant independent of the subdomain size and the mesh size. In this talk, we will discuss a further study for a dual iterative substructuring method with a penalty term in terms of its convergence analysis and practical efficiency. On one hand, the convergence studies in [2, 3] rule out the case when a relatively small η is taken. On the other hand, the condition number estimate without any size limitation on η will be presented. Moreover, based on the close relationship between the FETI-DP method and the proposed method, which results from the condition number estimate, it is shown that a penalty parameter chosen in a certain range, called a nearly optimal range of η, is sufficient to accelerate the convergence of the dual iteration. In addition, inner preconditioners for subdomain problems will be discussed for an improvement of practical efficiency.

References

[1] C. Farhat and M. Lesoinne and K. Pierson. A scalable dual-primal domain de- composition method. Numer. Linear Algebra Appl., 7 (2000), pp. 687–714.

[2] C.-O. Lee and E.-H. Park. A dual iterative substructuring method with a penalty term. Numer. Math., 112 (2009), pp. 89–113.

[3] C.-O. Lee and E.-H. Park. A dual iterative substructuring method with a penalty term in three dimensions. Comput. Math. Appl., 64 (2012), pp. 2787–2805.

2-15 2: Parallel session talks

THE PARTITION OF UNITY METHOD FOR THE 3D ELASTIC WAVE PROBLEMS IN THE HIGH FREQUENCY DOMAIN M. Mahmood1a, O. Laghrouche1b, A. El-Kacimi1c and J. Trevelyan2

1Institute for Infrastructure and Environment, Heriot-Watt University Edinburgh EH14 4AS, UK [email protected], [email protected], [email protected] 2School of Engineering and Computing Sciences, Durham University Durham DH1 3LE, UK [email protected]

Key words: PUFEM, finite element method, plane waves, elastic waves, 3D.

A growing research has been developed on wave numerical modelling in different fields e.g. in seismolog, geophysics, soil mechanics and biomedical ultrasound. The related problems are modeled using mainly Helmhotz, Maxwell’s and Navier’s elastic wave equations depending on wave propagation media and type of application. A number of different numerical methods have been used to solve the elastic wave equations but among the most commonly used is finite element method (FEM), due to e.g. its flexibility in handling complex geometries, its ability to handle different types of media etc.. However, the use of standard finite element method for solving acoustic or elastic problems, in medium and high frequency domain, becomes limited in terms of memory capacity and computationally very expensive in terms of CPU time. Our object is to develop finite elements, for three dimensional elastic wave problems, capable of containing many wavelengths per nodal spacing. This will be achieved by applying the plane wave basis decomposition to the 3D elastic wave equation. These elements will allow us to relax the traditional requirement of around ten nodal points per wavelength and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency. The accuracy and effectiveness of the proposed technique will be determined by comparing solutions for selected problems with available analytical solutions and/or to high resolution numerical solutions using conventional finite elements. The method of plane wave basis decomposition used to develop wave finite elements for the two-dimensional elastic wave equation [1-2] will be extended to three dimensions. The governing equation is a vector equation and multiple wave speeds are present for any given frequency. In an infinite elastic medium, there are two different types of wave propagating simultaneously, the dilatation or compression wave (P), and the distortional or shear wave (S). The application of the Helmholtz decomposition theorem to the displacement field yields a scalar wave equation for the P-wave potential and a vector wave equation for the S-wave potential. The two wave equations are independent but the boundary conditions depend on both P-wave and S-wave potentials, thus coupling the associated scalar P-wave and vector S-wave equations.

2-16 2: Parallel session talks

References

[1] El Kacimi A and Laghrouche O.: Numerical Modelling of Elastic Wave propagation in Frequency Domain by the Partition of Unity Finite Element Method. Interna- tional Journal for Numerical Methods in Engineering, 77: 1646–1669, 2009.

[2] El Kacimi A and Laghrouche O.: Improvement of PUFEM for the numerical so- lution of high frequency elastic wave scattering on unstructured triangular mesh grids. International Journal for Numerical Methods in Engineering, 84: 330–350, 2010.

2-17 2: Parallel session talks

ON THE NUMERICAL TREATMENT OF ESSENTIAL BOUNDARY CONDITIONS WITHIN POSITIVITY-PRESERVING FINITE ELEMENT METHODS FOR CONVECTION-DOMINATED TRANSPORT PROBLEMS Matthias M¨oller

Institute of Applied Mathematics (LS III), TU Dortmund University of Technology, Vogelpothsweg 87, D-44227 Dortmund, Germany [email protected]

The design of positivity-preserving numerical schemes for convection-dominated flow problems has been a topic of active research for many years. There is thus a large number of publications on stabilized finite element methods for the convection-diffusion equation such as SUPG, LPS or SOLD schemes. Their common aim is to reduce or at best prevent the generation of nonphysical undershoots and overshoots near steep gradients. While an improper implementation of essential boundary conditions may also cause spurious oscillations this aspect is hardly addressed in the literature. In this talk, the algebraic flux correction (AFC) methodology [2] is revisited and generalized to the case of weakly imposed essential boundary conditions [1]. In essence, the classical continuous Galerkin discretization of the convection-diffusion equation is converted into a non-oscillatory high-resolution scheme by adding artificial dissipation which is adaptively removed in regions where the solution is smooth with the aid of a flux limiter. All required information is extracted from the finite element system matrix which is manipulated based on rigorous mathematical constraints. We suggest a generalization of these positivity criteria which account for the additional contribu- tion of boundary integral terms to the discretized transport operator responsible for the weak imposition of boundary conditions. An alternative approach to prescribing Dirichlet boundary values (in strong sense) without violating positivity preservation was suggested in [3]. In this talk, both approaches are compared by presenting numer- ical examples for the two-dimensional convection-diffusion equation.

References

[1] Y. Bazilevs, T.J.R. Hughes: Weak imposition of Dirichlet boundary conditions in fluid mechanics. Computers & Fluids, 36(1), 2007, 1226.

[2] D. Kuzmin: Algebraic flux correction I. Scalar conservation laws. Chapter 6 in: D. Kuzmin, R. ¨ohner,S. Turek: Flux-Corrected Transport, Springer, 2nd edition 2012, 145–192.

[3] J. Niemeyer, B. Simeon: On finite element methodflux corrected transport stabilization for advection-diffusion problems in a par- tial differential-algebraic framework. Preprint. URN: http://nbn- resolving.de/urn/resolver.pl?urn:nbn:de:hbz:386-kluedo-34442, 2013.

2-18 2: Parallel session talks

ALGEBRAIC FLUX CORRECTION IN A PARTIAL DIFFERENTIAL-ALGEBRAIC FRAMEWORK Julia Niemeyera and Bernd Simeon

Felix-Klein Centre for Mathematics, University of Kaiserslautern, Paul-Ehrlich Straße 31, D-67663 Kaiserslautern, Germany, [email protected]

Applications like coupled multiphysics problems, domain decomposition or the magne- tohydrodynamic equations include constraints or invariants on the partial differential equation system. The partial differential-algebriac framework allows a general handling by appending these constraints by means of Lagrange multipliers. If an advection-dominated flow problem is part of the coupled system, the finite ele- ment method is known to tend to produce unphysical oscillations. Therefore a suitable stabilization method needs to be included. During the last decades a stabilization on the algebraic level by flux correction labeled as algebraic flux correction (AFC) has been introduced [1] and improved [2]. The main goal of this talk is the extension of the AFC method to the partial differential- algebraic framework. In short, to derive a stable numerical solution means to avoid birth and growth of unphysical local extrema on the one hand and to suppress global over- and undershoots on the other hand. While the prevention of unphysical local extrema can be derived by modifications of the system matrices, the used time inte- gration method needs to guarantee positivity preserving in every single timestep such that no global over- and undershoots arise. The numerical solution of a differential-algebraic equation (DAE), e.g., stemming from the semi-discretization of coupled problems with the finite element method, is not the same as for an ordinary differential equation (ODE). Particulary in the most general case the positivity preservation property of a time integrator in the ODE case cannot be transferred to the DAE case [3]. In this talk we concentrate on DAE systems stemming from a finite element discretiza- tion. After a brief introduction to DAEs we will show that the one-step θ−scheme remains positivity preserving when applied to a DAE under certain conditions. The Finite Element Method - Flux Corrected Transport algorithm will be modified to han- dle DAEs [4] and we show some first simulation results.

References

[1] D. Kuzmin, S. Turek: Flux correction tools for finite elements. J. Comput. Phys. 175, 525–558 (2002).

[2] D. Kuzmin: Algebraic flux correction I. Scalar conservation laws. Chapter 6 in: D. Kuzmin, R. Lohner, S. Turek: Flux-Corrected Transport, Springer, 2nd edition 2012, 145–192.

2-19 2: Parallel session talks

[3] A.-K. Baumann, V. Mehrmann: Numerical integration of positive linear differential-algebraic systems. Preprint 2012-02, Institut f¨ur Mathematik, TU Berlin, 2012

[4] J. Niemeyer, B. Simeon: On Finite Element MethodFlux Corrected Transport Stabilization for Advection-Diffusion Problems in a Partial Differential-Algebraic Framework. Preprint. URN: http://nbn-resolving.de/urn/resolver.pl?urn: nbn:de:hbz:386-kluedo-34442

2-20 2: Parallel session talks

APPROXIMATION OF EDDY CURRENTS IN AN AXISYMMETRIC UNBOUNDED DOMAIN Pilar Salgado1 and Virginia Selgas2

1Departamento de Matem´aticaAplicada, Escola Polit´ecnica Superior, Universidad de Santiago de Compostela, 27002 Lugo (Spain) [email protected] 2Departamento de Matem´aticas, Escuela Polit´ecnicade Ingenier´ıa, Universidad de Oviedo 33203 Gij´on(Spain) [email protected]

The time-harmonic eddy current model arises in physical and industrial applications such as the modeling of induction furnaces (see, for instance, [1]), where the physical geometry is axisymmetric and unbounded. To approximately solve this axisymmetric problem, we propose and analyze a symmetric FEM and BEM coupling method in terms of a magnetic vector potential. Moreover, we show that our formulation is well- posed, and also propose a discretization that leads to a convergent Galerkin scheme; see [3] and the references given therein. We finally develop some numerical techniques to implement this method in a MatLab code and show its applicability to simulate an industrial problem; see Fig. 1.

Fig. 1: Numerical results for the simulation of a small induction furnace composed by a graphite crucible containing silicon in its interior and a four-turns coil. Both figures show the approximation of the current density: the one on the left focus in the crucible, whereas that on the right shows the whole furnace.

References

[1] A. Berm´udez,D. G´omez,M.C. Mu˜niz,P. Salgado, R. V´azquez.Numerical simu- lation of a thermo-electromagneto-hydrodynamic problem in an induction heating furnace. Applied Numerical Mathematics 59 (2009), 2082-2104.

2-21 2: Parallel session talks

[2] A. Berm´udez,C. Reales, R. Rodr´ıguez,P. Salgado, Numerical analysis of a finite element method for the axisymmetric eddy current model of an induction furnace. IMA Journal of Numerical Analysis 30 (2010), 654-676.

[3] V. Selgas, A symmetric BEM-FEM method for an axisymmetric eddy current prob- lem. Submitted.

A UNIFORM CONVERGENCE ANALYSIS OF THREE-STEP TAYLOR GALERKIN FINITE ELEMENT MONOTONE ITERATIVE DOMAIN-DECOMPOSITION SCHEME FOR SINGULARLY PERTURBED PROBLEMS Vivek Sangwan1 and B. V. Rathish Kumar2

1School of Mathematics and Computer Applications, Thapar University, India, [email protected] 2Department of Mathematics and Statistics, IIT Kanpur, Kanpur, India. [email protected]

Singularly perturbed problems appear in almost all the branches of science and engi- neering. Singularly perturbed problems are characterized by a small parameter multi- plied by the highest order derivative terms. As this small parameter, generally called singularly perturbed parameter, approaches to zero, sharp boundary or internal layers evolve in the solution. Conventional methods fail to capture these layers. We need special techniques to capture these layers. Fitted mesh methods and fitted operator methods are two principle approaches which are used to find the approximate solution of singularly perturbed problems. We have used both the techniques in our study of approximate solutions of these problems. The present contribution concerns the sta- bility and uniform convergence results for a nonlinear singularly perturbed problem using three-step Taylor Galerkin finite element scheme via monotone iterative domain decomposition algorithm for nonoverlappping domains. Shishkin mesh has been used for domain discretization. The monotone convergence of the monotone iterative al- gorithm and monotone iterative domain decomposition algorithm has been proved by showing that the sequence of upper or lower solutions generated by the iterative scheme converges to the approximate solution of the discretized problem. Thus the uniform convergence of the proposed nonoverlapping domain decomposition scheme has been established under three-step Taylor Galerkin finite element framework and is shown to be of order ∆t3 + ∆x. Numerical experiments have been carried out to depict the robustness and efficiency of the proposed scheme in capturing very sharp boundary layers.

2-22 2: Parallel session talks

WITH A HIERARCHICAL ERROR INDICATOR TOWARD ANISOTROPIC MESH REFINEMENT Ren´eSchneider

TU Chemnitz, Fakult¨atf¨urMathematik, Chemnitz, Germany, [email protected]

We propose a new approach to adaptive mesh refinement. Instead of considering local mesh diameters and their adaption to solution features, we propose to evaluate the benefit of possible refinements in a direct fashion, and to select the most profitable refinements. The resulting refinement guide can be seen as a hierarchical edge-error indicator. We demonstrate that based on this approach a directional refinement of triangular elements can be achieved, allowing arbitrarily high aspect ratios. However, only with the help of edge swapping and/or node removal (directional un- refinement) near optimal performance can be achieved for strongly anisotropic solution features. With these ingredients even re-alignment of the mesh with arbitrary error directions is achieved. Numerical experiments demonstrate the utility of the proposed anisotropic refinement strategy.

COMPUTATIONAL ASPECTS IN SMOOTH APPROXIMATION OF DATA Karel Segeth

Technical University of Liberec, Liberec, Czech Republic [email protected]

In the seventies, Talmi and Gilat introduced a way of data approximation called smooth. Our concern in the paper is to show mathematical properties of this pro- cedure applied to the exact interpolation as well as to the fitting of data when, at the same time, we take into account the smoothness of the approximation curve and its derivatives. This curve is the solution of a variational problem with constraints that represent the interpolation conditions at nodes. Some procedures known in numerical analysis, e.g. spline approximation, can be considered a special case of the smooth approximation. In particular, we are concerned with the choice of the system exp(ikx) for the basis functions of the smooth approximation. This is the case when the evaluation of cer- tain infinite series needed in the approximation process can be transformed into the computation of Fourier integrals. We present results of some 1D numerical examples that show advantages and drawbacks of this procedure. This research has been supported by the Institute of Novel Technologies and Applied Informatics, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Tech- nical University of Liberec.

2-23 3: Mini-Symposium: A priori finite element error estimates in optimal control

3 Mini-Symposium: A priori finite element error estimates in optimal control

Organiser: Thomas Apel

3-1 3: Mini-Symposium: A priori finite element error estimates in optimal control

A PRIORI ERROR ESTIMATES FOR FINITE ELEMENT METHODS FOR H(2,1)-ELLIPTIC EQUATIONS Thomas Apel1a, Thomas G. Flaig1b and Serge Nicaise2

1Institut f¨urMathematik und Bauinformatik, Universit¨atder Bundeswehr M¨unchen, D-85579 Neubiberg, Germany [email protected] [email protected] 2LAMAV, Institut des Sciences et Techniques de Valenciennes, Universit´ede Valenciennes et du Hainaut Cambr´esis, B.P. 311, 59313 Valenciennes Cedex, France [email protected]

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In our talk we will introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other. We establish a regularity result and estimates for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use cubic Hermite elements, in the other dimension Lagrange elements of order k = 1, 2, 3. For these elements we prove the error bound O(h2 + τ k) in the energy norm and O (h2 + τ k)(h2 + τ)
in the L2(Q)-norm. This type of equations appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions.

3-2 3: Mini-Symposium: A priori finite element error estimates in optimal control

CRANK-NICOLSON AND STORMER-VERLET¨ DISCRETIZATION SCHEMES FOR OPTIMAL CONTROL PROBLEMS WITH PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS Thomas Apela and Thomas G. Flaigb

Institut f¨urMathematik und Bauinformatik, Universit¨atder Bundeswehr M¨unchen, 85577 Neubiberg, Germany [email protected], [email protected]

In this talk we present the discretization of optimal control problems with parabolic partial differential equations. In particular we discuss the problem

Z T Z T α 2 β 2 ν 2 min ky(·,T ) − yDkH + ky − ydkH dt + kukH dt 2 2 0 2 0 s.t. yt + Ay = u in Ω × (0,T ], ∂y = 0 on ∂Ω × (0,T ], ∂n y(·, 0) = v in Ω, with a second order elliptic operator A and H = L2(Ω). For the temporal discretization we focus on a family of Crank-Nicolson schemes with different discretizations for the state y and the adjoint state p so that discretization and optimization commute. One of the schemes can be explained as the St¨ormer- Verlet scheme. So we can interpret the method in the context of geometric numerical integration. The Hamiltonian structure of the parabolic optimal control problem is also discussed. Further we point out that two schemes may also be obtained as a Galerkin method with quadrature. If we use linear finite elements for the spatial discretization, we can prove second order convergence in space and time for one of the schemes. Finally we present numerical examples where second order convergence in time is observed.

References

[1] Thomas Apel and Thomas G. Flaig. Crank-Nicolson schemes for optimal con- trol problems with evolution equations. SIAM Journal on Numerical Analysis, 50(3):1484–1512, 2012.

[2] Thomas G. Flaig. Discretization strategies for optimal control problems with parabolic partial differential equations. PhD thesis, Universit¨atder Bundeswehr M¨unchen, submitted, 2013.

3-3 3: Mini-Symposium: A priori finite element error estimates in optimal control

ERROR ESTIMATES FOR DIRICHLET CONTROL PROBLEMS IN POLYGONAL DOMAINS Thomas Apel1, Mariano Mateos2, Johannes Pfefferer3 and Arnd R¨osch4

1Institut f¨urMathematik und Bauinformatik, Universit¨atder Bundeswehr M¨unchen, 85577 Neubiberg, Germany [email protected] 2Departamento de Matem´aticas,E.P.I. de Gij´on, Universidad de Oviedo, Campus de Gij´on, 33203 Gij´on,Spain [email protected] 3Institut f¨urMathematik und Bauinformatik, Universit¨atder Bundeswehr M¨unchen, 85577 Neubiberg, Germany [email protected] 4Fachbereich Mathematik, Universt¨atDuisburg-Essen, Forsthausweg 2, 47057 Duisburg, Germany [email protected]

We study a control constrained Dirichlet optimal control problem governed by an el- liptic equation posed on a domain with polygonal boundary. We admit both the cases of a convex or a non-convex domain. First, we make a detailed study of the regularity of the solution near the corners. Despite the non-convexity of the domain we are able to prove that the optimal control is a continuous function. We can even precise its regularity by using classical Sobolev spaces W 1,p(Γ), where p > 2 depends on the greatest convex angle of the domain. The regularity of the state and of the adjoint state is also studied, both in the framework of classical Sobolev spaces and of weighted Sobolev spaces. Finally, we obtain error estimates for the finite element approximation of the prob- lem. We discretize both control and state by means of continuous piecewise linear functions on quasi-uniform meshes. The order of convergence depends on the regular- ity of the data and on the angles of the domain. Some numerical experiments are presented to illustrate the theoretical estimates.

3-4 3: Mini-Symposium: A priori finite element error estimates in optimal control

BOUNDARY CONCENTRATED FEM FOR OPTIMAL CONTROL PROBLEMS Sven Beuchler

Institute for Numerical Simulation, University of Bonn, Bonn, Germany. [email protected]

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite ∗ ∗ element method. We prove that the discretization error ku − uhkL2(Γ) decreases like N −1, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the dis- cretization error behaves like N −3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The talk is complemented with numerical results. The presentation is the result of collaborations with K. Hofer (Bonn), D. Wachsmuth, J.-E. Wurst (Wuerzburg).

ERROR ESTIMATES FOR THE VELOCITY TRACKING PROBLEM USING DUALITY ARGUMENTS Konstantinos Chrysafinos

Department of Mathematics, National Technical University of Athens, Athens, Greece. [email protected],

An optimal control problem related to the velocity tracking problem of Navier-Stokes flows is considered. The main goal is to minimize the tracking functional using dis- tributed controls with control constraints. The schemes under consideration are based on a discontinuous time-stepping approach combined with standard conforming finite elements for the spacial discretization. Special emphasis will be placed to the role of duality arguments in developing error estimates, and to the role of the related adjoint equation which exhibits very interesting structural properties. This is a joint work with Eduardo Casas, University of Cantabria, Santander, Spain.

3-5 3: Mini-Symposium: A priori finite element error estimates in optimal control

CONVERGENCE AND ERROR ANALYSIS OF A NUMERICAL METHOD FOR THE IDENTIFICATION OF MATRIX PARAMETERS IN ELLIPTIC PDES Klaus Deckelnick1 and Michael Hinze2

1Institut f¨urAnalysis und Numerik, Otto–von–Guericke–Universit¨atMagdeburg, Universit¨atsplatz2, 39106 Magdeburg, Germany [email protected] 2Schwerpunkt Optimierung und Approximation, Universit¨atHamburg, Bundesstraße 55, 20146 Hamburg, Germany [email protected]

In this talk we present and analyze a numerical method for solving the inverse problem of identifying the diffusion matrix in an elliptic PDE from distributed noisy measure- ments. We use a regularized least squares approach in which the state equations are given by a finite element discretization of the elliptic PDE. The unknown matrix param- eters act as control variables and are handled with the help of variational discretization as introduced in [M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30, 45–61 (2005)]. For a suitable coupling of Tikhonov regularization parameter, finite element grid size and noise level we are able to prove L2–convergence of the discrete solutions to the unique norm–minimal solution of the identification problem; corresponding convergence rates can be obtained provided that a suitable projected source condition is fulfilled. Finally, we present a numerical experiment which supports our theoretical findings.

3-6 3: Mini-Symposium: A priori finite element error estimates in optimal control

OPTIMAL CONTROL OF BIHARMONIC OPERATOR Stefan Frei1a, Rolf Rannacher1b and Winnifried Wollner2

1University of Heidelberg, Department of Mathematics, INF 293/294, 69120 Heidelberg, Germany [email protected], [email protected] 2University of Hamburg, Department of Mathematics, Bundesstr 55, 20146 Hamburg, Germany [email protected]

In this talk a priori error estimates are derived for the finite element discretization of optimal distributed control problems governed by the biharmonic operator. The state equation is discretized in primal mixed form using continuous piecewise biquadratic finite elements, while piecewise constant approximations are used for the control. The error estimates derived for the state variable as well as that for the control are order- optimal on general unstructured meshes. However, on uniform meshes not all error estimates are optimal due to the low-order control approximation. All theoretical results are confirmed by numerical tests.

AN INTERIOR PENALTY METHOD FOR DISTRIBUTED OPTIMAL CONTROL PROBLEMS GOVERNED BY THE BIHARMONIC OPERATOR Thirupathi Gudi, Neela Nataraja and Veeranjaneyulu Sadhanala

Department of Mathematics, IIT Bombay, Powai, Mumbai 400076. India [email protected] http://www.math.iitb.ac.in/~neela

In the past few years, C0 interior penalty methods have been attractive for solving the fourth order problems. In this work, a C0 interior penalty method has been proposed and analyzed for an optimal control problem governed by the biharmonic operator. The state equation is discretized using continuous piecewise quadratic finite elements while piecewise constant approximations are used for the control variable. Error estimates are derived for both the state and control variables. Theoretical results are demonstrated by numerical experiments.

3-7 3: Mini-Symposium: A priori finite element error estimates in optimal control

A PRIORI ERROR ESTIMATES FOR PARABOLIC OPTIMAL CONTROL PROBLEMS WITH POINT CONTROLS Dmitriy Leykekhman1 and Boris Vexler2

1Department of Mathematics, University of Connecticut, USA. [email protected] 2Technische Universit¨atM¨unchen, Faculty of Mathematics, Boltzmannstraße 3, 85748 Garching b. Munich, Germany [email protected]

In this talk, we consider the following optimal control problem:

Z T Z T 1 2 α 2 min J(q, u) := ku(t) − ub(t)kL2(Ω)dt + |q(t)| dt q,u 2 0 2 0 subject to the second order parabolic equation

ut(t, x) − ∆u(t, x) = q(t)δx0 , (t, x) ∈ I × Ω, u(t, x) = 0, (t, x) ∈ I × ∂Ω, u(0, x) = 0, x ∈ Ω and subject to pointwise control constraints

qa ≤ q(t) ≤ qb a. e. in I.

2 Here I = [0,T ], Ω ⊂ R is a convex polygonal domain, x0 ∈ IntΩ fixed, and δx0 is the Dirac delta function. The parameter α is assumed to be positive and the desired 2 ∞ state ub fulfills ub ∈ L (I; L (Ω)). The control bounds qa, qb ∈ R ∪ {±∞} fulfill qa < qb. Such problems are challenging due to low regularity of the state equation. We use the standard continuous piecewise linear approximation in space and the first order discontinuous Galerkin method in time to approximate the problem numerically. De- spite low regularity of the state equation, we obtain almost optimal h2 + k convergence rate for the control in L2 norm, without any relationship between the size of the space discretization h and the time steps k. The main ingredients of the analysis are sharp regularity results and the new global and local fully discrete a priori pointwise in space and L2 in time error estimates the parabolic problems.

3-8 3: Mini-Symposium: A priori finite element error estimates in optimal control

OPTIMAL ERROR ESTIMATES FOR FINITE ELEMENT DISCRETIZATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH FINITELY MANY POINTWISE STATE CONSTRAINTS Dmitriy Leykekhman1, Dominik Meidner2a and Boris Vexler2b

1University of Connecticut, Department of Mathematics, 196 Auditorium Road, Storrs, CT 06269-3009,USA [email protected] 2Chair of Optimal Control, Technische Universit¨atM¨unchen, Faculty of Mathematics, Boltzmannstraße 3, 85748 Garching b. Munich, Germany [email protected], [email protected]

In this talk, we consider the following model elliptic optimal control problem with finitely many state constraints on a convex smooth domain Ω in two and three dimen- sions:

1 2 α 2 Minimize ku − u k 2 + kqk 2 2 d L (Ω) 2 L (Ω) subject to the state equation

−∆u = q in Ω, u = 0 on ∂Ω and to finitely many pointwise state constraints

u(xi) = bi i = 1, 2, . . . , n,

Here, x1, x2, . . . , xn ∈ Ω are mutually distinct given points. Such problems are challenging due to low regularity of the adjoint variable. For the discretization of the problem we consider continuous linear elements on quasi- uniform and graded meshes separately. Our main result establishes optimal a priori error estimates for the control, state, and the Lagrange multiplier on the two mentioned types of meshes, see following table.

Proved orders of convergence on quasi-uniform / graded meshes

Dimension Error in control Error in state Error in Lagrange multiplier 2 1 / 2 2 / 2 2 / 2 1 3 2 / 2 1 / 2 1 / 2

In particular, in three dimensions the optimal second order convergence rate for all three variables is possible only on properly refined graded meshes. The derived estimates are illustrated by a numerical example.

3-9 3: Mini-Symposium: A priori finite element error estimates in optimal control

VERIFICATION OF OPTIMALITY CONDITIONS AND DISCRETIZATION ERROR ESTIMATES Martin Naßa and Arnd R¨oschb

Faculty of Mathematics, University of Duisburg-Essen, Germany. [email protected], [email protected]

Optimal control of a semilinear elliptic partial differential equation is a nonconvex op- timization problem. Hence second-order sufficient conditions are needed to ensure local optimality. Such conditions allow to derive a priori error estimates for FE- discretizations. However, this strategy has an essential drawback. The second-order condition has to be satisfied in the exact solution, but only a numerical approximation of the exact solution is available. Consequently it is impossible to check the second-order sufficient condition. In this talk we present another strategy. We require only a coercivity condition for the numerical solution which can be checked numerically. This is the main tool to show discretization error estimates for a FE-discretization.

ON DISCRETIZED NONCONVEX ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH POINTWISE STATE CONSTRAINTS Ira Neitzel1, Johannes Pfefferer2 and Arnd R¨osch3

1 Technische Universit¨atM¨unchen Centre for Mathematical Sciences, M17 Boltzmannstr. 3 D-85748 Garching b. Munich, Germany [email protected] 2 Universit¨atder Bundeswehr M¨unchen, Institut f¨urMathematik und Bauinformatik Fakult¨atf¨urBauingenieur- und Vermessungswesen D-85577 Neubiberg, Germany [email protected] 3 Universit¨atDuisburg-Essen, Fachbereich Mathematik, Forsthausweg 2 D-47057 Duisburg, Germany [email protected]

We discuss properties of finite-element-discretized optimal control problems subject to pointwise state constraints and a semilinear elliptic state equation. We focus on two aspects of the discretized problem. First, we discuss the convergence of discrete locally optimal controls to associated solutions of the continuous problem. We prove a rate of convergence depending on the discretization parameter h with the help of second order sufficient conditions (SSC) for the original problem. Second, we discuss how these SSC for the continuous problem transfer to the discrete level. This is motivated by the fact that for instance the proof of convergence of the SQP method or properties like local uniqueness of solutions rely on SSC.

3-10 3: Mini-Symposium: A priori finite element error estimates in optimal control

SPARSE ELLIPTIC CONTROL PROBLEMS IN MEASURE SPACES: REGULARITY AND FEM DISCRETIZATION Konstantin Piepera and Boris Vexlerb

Chair of Optimal Control, Technische Universit¨atM¨unchen, Faculty of Mathematics, Boltzmannstraße 3, 85748 Garching b. Munich, Germany [email protected] [email protected]

In this talk we consider an optimal control problem governed by an elliptic equation, where the control variable lies in a measure space. This formulation leads to a sparse structure of the optimal control, which provides among other things an elegant way to attack problems of optimal source placement. We discuss the functional analytic setting of the problem under consideration and the regularity issues of the optimal solution. Moreover, we present a discretization concept and prove a priori error estimates for the discretization error, which significantly improve the estimates from the literature. Numerical examples for problems in two and three space dimensions illustrate our results.

OPTIMAL BOUNDARY CONTROL PROBLEMS IN ENERGY SPACES Olaf Steinbach

Institut f¨urNumerische Mathematik, TU Graz, Steyrergasse 30, A 8010 Graz, Austria [email protected]

We consider Dirichlet boundary control problems subject to second order partial differ- ential equations where the set of admissible functions is a subset of the related energy space, i.e. of H1/2(Γ). An equivalent norm is induced by the Steklov–Poincar´eoper- ator which realizes the Dirichlet to Neumann map for a given Dirichlet control. The reduced formulation results in a variational inequality for the unknown control, which is equivalent to a partial differential equation with bilateral constraints of Signorini type on the boundary. We discuss finite and boundary element discretizations of the optimality system and we present related a priori error estimates both in the energy 1/2 norm in H (Γ) and in L2(Γ). Applications involve the stationary and transient heat equation, and the boundary control of the stationary Navier–Stokes equations. This talk is based on joint work with L. John, A. Kimeswenger, G. Of, and T. X. Phan.

3-11 3: Mini-Symposium: A priori finite element error estimates in optimal control

FINITE ELEMENT METHODS FOR FOURTH ORDER VARIATIONAL INEQUALITIES ARISING FROM ELLIPTIC OPTIMAL CONTROL PROBLEMS Li-yeng Sung

Department of Mathematics and Center for Computation and Technology, Louisiana State University, LA 70803, USA. [email protected]

In this talk we will discuss finite element methods for elliptic optimal control problems with pointwise state constraints formulated as fourth order variational inequalities. This is joint work with S.C. Brenner and Y. Zhang.

3-12 4: Mini-Symposium: Analysis and applications of boundary element methods

4 Mini-Symposium: Analysis and applications of boundary element methods

Organisers: Martin Schanz and Olaf Stein- bach

4-1 4: Mini-Symposium: Analysis and applications of boundary element methods

THE BEM++ BOUNDARY ELEMENT LIBRARY AND APPLICATIONS S.R. Arridge1a, T. Betcke2b, M. Schweiger1c and W. Smigaj´ 2d

1Department of Computer Science, University College London, UK [email protected], [email protected], 2Department of Mathematics, University College London, UK [email protected], [email protected]

BEM++ is an open source Galerkin boundary element library for the solution of sys- tems of boundary integral equations. Its core is developed in C++ with a convenient Python interface on top of it. It offers support for the fast assembly and approxi- mate LU decomposition of boundary integral operators via hierarchical matrices, and connects to the Trilinos library for iterative solvers. Currently, BEM++ supports Laplace, Helmholtz, and Maxwell problems. In this talk we give an introduction to the library, and demonstrate various application ex- amples, including layered problems in optical tomography, and convolution quadrature methods for time-domain problems.

4-2 4: Mini-Symposium: Analysis and applications of boundary element methods

A RECURSIVE INTEGRAL EQUATIONS APPROACH FOR ELECTROMAGNETIC SCATTERING BY BIPERIODIC MULTILAYER GRATINGS Beatrice Bugert1,2,a and Gunther Schmidt1

1 Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany 2 Berlin Mathematical School, Technical University Berlin, Berlin, Germany a [email protected]

Scattering theory has numerous applications in micro-optics like the construction of holographic films, optical storage disks and antireflective coatings. Many of these optical devices are implemented by a multilayered structure. We study the special case of electromagnetic scattering by biperiodic multilayered structures modeled by a finite number of smooth, non-selfintersecting surfaces Σj, j = 0,...,N, which are 3 2π-periodic in both x1- and x2-direction and separate regions Gj ⊂ R filled with materials of constant electric permittivity j and constant magnetic permeability µj. i The scattering of a time-harmonic plane wave E incident on the top layer Σ0 of the multilayered structure from G0 is computed by solving

2 curl curl Ej − κ Ej = 0 in G N , j j∈J0 i

−1 −1 i

nj × E1 − E0 − E = 0, nj × µ1 E1 − µ0 E0 + E = 0 on Σ0, −1 −1
nj × (Ej+1 − Ej) = 0, nj × µj+1Ej+1 − µj Ej = 0 on Σj∈J ,

N where J = {1,...,N − 1}, J0 = J ∪ {0,N}, and, additionally making sure that the outgoing wave condition at infinity is satisfied. For the study of the above electromag- netic scattering problem, we propose a recursive integral equation algorithm following the scheme of [3], in which the equivalent problem for oneperiodic multilayered struc- tures was treated. The combined use of a Stratton-Chu integral representation and an electric potential ansatz yields a singular integral equation on each interface. These equations arise from eachother via recursion from the bottom to the top interface lead- ing to a recursive algorithm. The idea for this potential ansatz is taken from [2]. We establish necessary and sufficient conditions such that the existence of solutions arising from the recursive integral equation algorithm imply that the electromagnetic scatter- ing problem is solvable. This result follows in particular from the Fredholm properties of one of the occuring operators in the algorithm, which can be established using the techniques in [1] and the results in [2]. Following [4], it is also possible to find conditions ensuring the uniqueness of solutions of the electromagnetic scattering problem.

References

[1] T. Arens, Scattering by biperiodic layered media: the integral equation approach, Habilitation thesis, KIT, Karlsruhe, 2010. [2] M. Costabel and F. Le Lou¨er, On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body, SIAM J. Appl. Math., 71 (2011),

4-3 4: Mini-Symposium: Analysis and applications of boundary element methods

pp. 635–656.

[3] G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equa- tions approach, Preprint of the Weierstrass Institute, 1601 (2011).

[4] G. Schmidt, Integral methods for conical diffraction, Preprint of the Weierstrass Institute, 1435 (2009).

BLACK-BOX PRECONDITIONING OF FEM/BEM MATRICES BY H-MATRIX TECHNIQUES Markus Faustmanna, Jens Markus Melenkb and Dirk Praetoriusc

Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria. [email protected], [email protected], [email protected]

Various compression techniques for matrices such as H-matrices have been developed in the past to store dense matrices and realize the matrix-vector-multiplication with log-linear (or even linear) complexity. One particular strength of H-matrices is that the H-format includes some arithmetics that provides the (approximate) inversion, (approximate) LU-decomposition etc. In our talk, we establish that the H-matrix format is rich enough to permit good approximations of inverse FEM (for various boundary conditions) and BEM matrices. The question of approximating the inverses of system matrices in the H-format by approximate H-inversion or H-LU factorization has previously, due to the method of proof, only been studied in the context of FEM and for Dirichlet boundary conditions [3, 4]. A main feature of our analysis is that we work in a fully discrete setting and thus avoid any additional projection errors. Therefore, we get approximations of arbitrary accuracy, and we show exponential convergence in the block rank. One possible application is the black-box preconditioning of the FEM/BEM systems in iterative solvers by use of an H-LU decomposition. Numerically it has been observed that such an approach works well in practice [5]. In our talk, we give a mathematical underpinning to these observations. Moreover, we consider matrices arising from the FEM-BEM coupling. Our results for the FEM case (stabilized Neumann problem) and the BEM case (simple-layer op- erator) can be used to derive an efficient block diagonal preconditioner that is based on the H-LU decomposition in the corresponding blocks.

References

[1] M.Faustmann, J.M.Melenk, D.Praetorius: H-matrix approximability of the inverses of FEM matrices, work in progress.

[2] M.Faustmann, J.M.Melenk, D.Praetorius: Existence of H-matrix approximants to the inverse of BEM matrices, work in progress.

4-4 4: Mini-Symposium: Analysis and applications of boundary element methods

[3] M. Bebendorf, W. Hackbusch: Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with L∞-coefficients, Numer. Math., 95 (2003), 1– 28.

[4] S. B¨orm:Approximation of solution operators of elliptic partial differential equa- tions by H- and H2-matrices. Numer. Math., 115 (2010), 165–193.

[5] L. Grasedyck: Adaptive Recompression of H-matrices for BEM, Computing, 74 (2005), 205–223.

ONE-EQUATION FEM-BEM COUPLING FOR ELASTICITY PROBLEMS Michael Feischl1, Thomas F¨uhrer1, Michael Karkulik2 and Dirk Praetorius1

1Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria [email protected] 2Faculdad de Mathem´aticas,Pontificia Unversidad Cat´olica de Chile

We consider a transmission problem in elasticity with a nonlinear material behavior in the bounded interior domain, which can be rewritten by means of the symmetric coupling as well as non-symmetric one-equation coupling methods, such as the Johnson- N´ed´eleccoupling. Problems arise when trying to prove solvability of the Galerkin discretization, because the space of rigid body motions is contained in the kernel of the Lam´eoperator. In this talk, which is based on the recent preprint [3], we present how to extend the ideas of implicit stabilization, developed for Laplace-type transmission problems in [1], to elasticity problems. We introduce modified equations which are fully equivalent (at the continuous as well as at the discrete level) to the original formulations. Solvability of the discrete modified problems, however, hinges on a condition on the discretization space, which states that the space is rich enough to tackle the rigid body motions. We prove that this condition is satisfied for regular triangulations, if the boundary element space contains the piecewise constants. Our analysis extends the works [2, 4, 5, 6]. Unlike [2], we avoid any assumption on the mesh-size. Unlike [4], we avoid the use of an interior Dirichlet boundary. Unlike [5, 6], we avoid any pre- and postprocessing steps as well as the numerical solution of additional boundary value problems.

References

[1] Aurada, Feischl, F¨uhrer,Karkulik, Melenk, Praetorius: Classical FEM-BEM cou- pling methods: nonlinearities, well-posedness, and adaptivity. Comput. Mech., pub- lished online first, (2012).

[2] Carstensen, Funken, Stephan: On the adaptive coupling of FEM and BEM in 2d- elasticity. Numer. Math., 77 (2012), 187–221.

4-5 4: Mini-Symposium: Analysis and applications of boundary element methods

[3] Feischl, F¨uhrer,Karkulik, Praetorius: Stability of symmetric and nonsymmetric FEM-BEM couplings in nonlinear elasticty. ASC Report 52, TU Wien (2012).

[4] Gatica, Hsiao, Sayas: Relaxing the hypotheses of Bielak-MacCamy’s BEM-FEM coupling. Numer. Math., 120 (2012), 465–487.

[5] Of, Steinbach: Is the one-equation coupling of finite and boundary element methods always stable? Berichte aus dem Institut f¨urNumerische Mathematik, TU Graz, 6 (2011).

[6] Steinbach: On the stability of the non-symmetric bem/fem coupling in linear elas- ticity. Comput. Mech., published online first, (2012).

4-6 4: Mini-Symposium: Analysis and applications of boundary element methods

AN AXIOMATIC APPROACH TO OPTIMALITY OF ADAPTIVE ALGORITHMS WITH APPLICATIONS TO BEM Michael Feischla and Dirk Praetoriusb

Vienna University of Technology, Institute for Analysis and Scientific Computing, Vienna, Austria [email protected], [email protected]

Based on recent joint work [Carstensen-Feischl-Page-Praetorius, 2013+], we consider abstract h-adaptive algorithms of the form

solve −→ estimate −→ mark −→ refine

Given a triangulation T , we assume that we can compute some approximation UT of the exact solution u. Starting from an initial triangulation T0 and given an accuracy ε > 0, the adaptive algorithm aims to iteratively refine a minimal number of elements of T0, T1, T2,... such that for some N > 0 it holds

ku − UTN k ≤ ε.

Analyzing the existing literature on h-adaptive algorithms, we extract a set of axioms which are sufficient —and in some cases even necessary— to obtain convergence with quasi-optimal rates

−s ku − UT k ≤ C(#T ) , with s > 0 as large as theoretically allowed by the structure of the problem (e.g. s = 1/d for lowest-order FEM for the Poisson problem in Rd). The abstract analysis developed is then employed to study applied problems like the weakly- as well as the hyper-singular integral equation

V u = (1/2 + K)g resp. W u = (1/2 − K0)ψ on Γ, where the approximations UT are computed via the boundary element method (BEM) and the adaptive algorithm is driven by some weighted residual error estimator η` from [Carstensen-Maischak-Stephan, 2000] (see [Feischl-Karkulik-Melenk-Praetorius, 2013]). Using the abstract results, we are able to prove quasi-optimal convergence rates for the BEM of both equations. Since the abstract approach applies also to the finite element method (FEM), our next step is to study the coupling of FEM and BEM.

4-7 4: Mini-Symposium: Analysis and applications of boundary element methods

A REDUCED BASIS BOUNDARY ELEMENT METHOD FOR A CLASS OF PARAMETERIZED ELECTROMAGNETIC SCATTERING MODEL M. Ganesh1, J. S. Hesthaven2 and B. Stamm3

1 Colorado School of Mines, Golden, CO 80401, USA. [email protected] 2 Brown University, Providence, RI 02912, USA. [email protected] 3 UPMC University of Paris 06 and CNRS France. [email protected]

We consider a parameterized multiple scattering wave propagation model in three di- mensions. The parameters in the model describe the location, orientation, size, shape, and number of scattering particles as well as properties of the input source field such as the frequency, polarization, and incident direction. The need for fast and efficient (online) simulation of the interacting scattered fields under parametric variation of the multiple particle surface scattering configuration is fundamental to several applications for design, detection, or uncertainty quantification. For such dynamic parameterized multiple scattering models, the standard dis- cretization procedures are prohibitively expensive due to the computational cost as- sociated with solving the full model for each online parameter choice. In this work, we propose an iterative offline/online reduced basis approach for a boundary element method to simulate a parameterized system of surface integral equations reformulation of the multiple particle wave propagation model. The approach includes (i) a greedy algorithm based computationally intensive offline procedure to create a selection of a set of a snapshot parameters and the construction of an associated reduced boundary element basis for each reference scatterer and (ii) an inexpensive online algorithm to generate the surface current and scattered field of the parameterized multiple wave propagation model for any choice of parameters within the parameter domains used in the offline procedure. Comparison of our numerical results with experimentally measured results for some benchmark configuration demonstrate the power of our method to rapidly simulate the interaction of scattered wave fields under parametric variation of the overall multiple particle configuration.

4-8 4: Mini-Symposium: Analysis and applications of boundary element methods

RETARDED POTENTIAL BOUNDARY INTEGRAL EQUATIONS FOR SOUND RADIATION IN A HALF-SPACE Heiko Gimperlein

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen O, Denmark, [email protected]

Motivated by the sound radiation of tires, we discuss a time-dependent boundary in- 3 tegral formulation for the wave equation in R+ outside a bound domain Ω. Using a 3 Green’s function which satisfies the acoustic boundary conditions on ∂R+, the exte- 3 rior problem reduces to an integral equation on ∂Ω ∩ R+. It is solved by Galerkin approximation and analytical convolution in time. We investigate the properties of the relevant boundary integral operators and deduce a priori and a posteriori error estimates for the numerical solution. Also computational aspects will be considered.

ANALYSIS OF A NON-SYMMETRIC COUPLING OF INTERIOR PENALTY DG AND BEM Norbert Heuer1 and Francisco-Javier Sayas2

1Facultad de Matem´aticas,Pontificia Universidad Cat´olicade Chile, Santiago, Chile [email protected] 2University of Delaware, Newark, Delaware, USA [email protected]

We present an analysis of a non-symmetric coupling of interior penalty discontinuous Galerkin and boundary element methods in two and three dimensions. Main results are discrete coercivity of the method, and thus unique solvability, and quasi-optimal convergence. The proof of coercivity is based on a localized variant of the variational technique from Sayas. This localization gives rise to terms which are carefully an- alyzed in fractional order Sobolev spaces, and by using scaling arguments for rigid transformations. Heuer acknowledges support by FONDECYT project 1110324.

4-9 4: Mini-Symposium: Analysis and applications of boundary element methods

ADAPTIVE NONCONFORMING BOUNDARY ELEMENT METHODS Norbert Heuera and Michael Karkulikb

Facultad de Matem´aticas, Pontificia Universidad Cat´olicade Chile, Santiago, Chile. [email protected], [email protected]

Nonconforming approximations to functions can offer various advantages. Usually, non- conforming discretizations for the approximate solution of PDE drop certain continuity requirements, which makes them flexible as well as easier to implement than their con- forming counterparts. For example, the approach by M. Crouzeix and P.-A. Raviart [1] uses discontinuous basis functions, and inter-element continuity is imposed weakly by enforcing edge or face jumps to have vanishing integral mean. This approach, and var- ious others, have been developed and analyzed rigorously for finite element methods. The situation is less developed in BEM. In [3], the approach of Crouzeix and Raviart is employed for solving the Laplacian’s hypersingular integral equation

W u = f on a screen Γ. It is shown that uniform mesh refinement yields a convergence rate O(h1/2) if u ∈ H1(Γ). The aim of this talk is to present first extensions of this approach, regarding adap- tivity. First, we present numerical examples which lead to the conjecture that, contrary as to usual expectations, imposing a higher regularity on u does not increase the con- vergence rate for uniform refinement. According to the Strang lemma, this is due to the consistency error, and we discuss briefly why it cannot be expected to converge with a higher rate. Hence, the optimal order for this method seems to be O(h1/2). However, if the solution has a reduced regularity, i.e. u ∈ Hs(Γ) for 1/2 < s < 1, the convergence rate for uniform refinement is reduced to O(hs−1/2). The second aim of our talk is to present an a posteriori error estimator, based on the h−h/2 methodology [2], which can be employed in an adaptive algorithm. Numerical examples indicate that adaptivity recovers the optimal rate.

References

[1] M. Crouzeix, P.-A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Fran¸caiseAutomat. In- format. Recherche Op´erationelleS´er.Rouge 7 (1973), 33–75.

[2] S. Ferraz-Leite, D. Praetorius: Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), 135–162.

[3] N. Heuer, F. J. Sayas: Crouzeix-Raviart boundary elements, Numer. Math. 112 (2009), 381–401.

4-10 4: Mini-Symposium: Analysis and applications of boundary element methods

PARALLEL BEM-BASED METHODS Dalibor Luk´aˇsa, Michal Mertab, Luk´aˇsMal´yc, Petr Kov´aˇrd and Tereza Kov´aˇrov´ae

Department of Applied Mathematics & IT4Innovations, VSB-Technicalˇ University of Ostrava, Czech Republic. [email protected], [email protected], [email protected], [email protected], [email protected]

In the first part of our contribution we propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differ- ential equations. In our method the underlying boundary element mesh is decomposed into N submeshes. Then the related N × N submatrices are assigned to N concurrent processes to be assembled. We additionally require each process to hold exactly one di- agonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and submatrices by cyclic decompositions of undirected complete graphs. The resulting algorithm√ enjoys parallel computational scalability O(1/N) and memory scalability O(1/ N). This is documented by numerical experiments up to 3 millions of boundary elements and 133 processors. Hierarchical matrices with the adaptive cross approximation as well as the fast multipole method are employed. In the second part a boundary-element counterpart of the domain decomposition vertex solver is proposed and tested for a 2-dimensional Poisson’s equation. While the standard theory has been developed only for triangular or quadrilateral subdomains, where harmonic base functions are available, the practical mesh-partitioners generate complex polygonal subdomains. We aim at bridging this gap. We construct the coarse solver on general polygonal subdomains so that the local coarse stifness matrix is approximated by a boundary-element discretization of the Steklov-Poincar´eoperator. The efficiency of our approach is documented by a substructuring into L-shape domains.

4-11 4: Mini-Symposium: Analysis and applications of boundary element methods

ON THE QUASI-OPTIMAL CONVERGENCE IN FEM-BEM COUPLING Jens Markus Melenk1a, Dirk Praetorius1b and B. Wohlmuth2

1Institut f¨urAnalysis und Scientific Computing, Technische Universit¨atWien, Austria Wiedner Hauptstr. 8-10, A-1040 Wien [email protected], [email protected] 2M2 Zentrum Mathematik Technische Universit¨atM¨unchen, Boltzmannstr. 3, D-85748 Garching, Germany [email protected]

We consider the symmetric coupling of FEM and BEM, which involves two field vari- ables, namely, one variable defined in the volume Ω and one on its boundary ∂Ω; the latter represents the exterior domain. The classical convergence theory of symmetric FEM-BEM coupling studies their joint approximation and shows a best approximation property for this joint approximation. However, typical discretizations feature differ- ent approximation powers for the two field variables: The approximation power of the space used to discretize the boundary variable is often better by h1/2 than that for the volume variable. In this talk, we show that this better approximation power of the space for the boundary variable can translate into better convergence rates for the boundary variable under suitable regularity assumptions.

4-12 4: Mini-Symposium: Analysis and applications of boundary element methods

ON THE ELLIPTICITY OF COUPLED FINITE ELEMENT AND ONE-EQUATION BOUNDARY ELEMENT METHODS FOR BOUNDARY VALUE PROBLEMS G¨unther Ofa and Olaf Steinbachb

Institute of Computational Mathematics, Graz University of Technology, Graz, Austria. [email protected], [email protected]

We present the extension of recent results on the stability of the Johnson–N´ed´eleccou- pling of finite and boundary element methods to the case of boundary value problems. In [1, 2, 3] the case of a free–space transmission problem was considered, and suf- ficient and necessary conditions are stated which ensure the ellipticity of the bilinear form for the coupled problem. The proof was based on the relation of the energies which are related to both the interior and exterior problem. When considering boundary value problems for both interior and exterior problems, additional estimates to bound the energy for the solutions of related subproblems are required. Moreover, several techniques for the stabilization of the coupled formulations are analyzed. Applications involve boundary value problems with either hard or soft inclusions, exterior boundary value problems, and macro–element techniques.

References

[1] G. Of, O. Steinbach. Is the one–equation coupling of finite and boundary element methods always stable? ZAMM - Journal of Applied Mathematics and Mechanics (2013) published online.

[2] F.–J. Sayas. The validity of Johnson–N´ed´elec’sBEM–FEM coupling on polygonal interfaces. SIAM J. Numer. Anal. 47 (2009) 3451–3463.

[3] O. Steinbach. A note on the stable one–equation coupling of finite and boundary elements. SIAM J. Numer. Anal. 49 (2011) 1521–1531.

4-13 4: Mini-Symposium: Analysis and applications of boundary element methods

RADIAL BASIS FUNCTIONS WITH APPLICATIONS TO ELASTICITY Sergej Rjasanowa and Richards Grzhibovskisb

Department of Mathematics, Saarland University, Germany [email protected], [email protected]

Radial basis functions (RBFs) have become increasingly popular for the construction of smooth interpolant s : Rn → R through a set o f N scattered, pairwise distinct data points. In the first part of the talk we introduce the RBF’s [1] and discuss their properties. The second part of the talk is devoted to the reconstruction of the three- dimensional metal sheet surfaces obtained via incremental forming techniques. [2]. In this application, the data comes from optical measurements of sheet metal parts. The top and the bottom surfaces of the part are measured in a fixed frame of reference, and a distribution of thickness along the part is sought. In the third part of the talk, a boundary integral formulation for a mixed boundary value problem in linear elasto- statics with a conservative right hand side is considered [3]. A meshless interpolant for the scalar potential of the volume force density is constructed by means of radial basis functions. An exact particular solution to the Lam´esystem with the gradient of this interpolant as the right hand side is found. Thus, the need of approximating the Newton potential is eliminated. The procedure is illustrated on numerical examples.

References

[1] H. Wendland. Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2005.

[2] M. Bambach, R. Grzhibovskis, G. Hirt, and S. Rjasanow. Adaptive cross ap- proximation for surface reconstruction based on radial basis functions. Journal of Engineering Mathematics, 62:149–160, 2008.

[3] H. Andr¨a,R. Grzhibovskis, and S. Rjasanow. Boundary Element Method for linear elasticity with conservative body forces. In T. Apel and O. Steinbach, editors, Advanced Finite Element Methods and Applications, number 66 in Lecture Notes in Applied and Computational Mechanics, pages 275–297. Springer-Verlag, Berlin- Heidelberg-NewYork, 2012.

4-14 4: Mini-Symposium: Analysis and applications of boundary element methods

STOKES FLOW ABOUT A COLLECTION OF SLIP SOLID PARTICLES A. Sellier

LadHyx. Ecole polytechnique, 91128 Palaiseau C´edex,France [email protected]

We consider a collection of N ≥ 1 arbitrarily-shaped solid slip particles immersed in a Newtonian liquid with uniform viscosity µ > 0 and density ρ > 0. Each particle Pn with center of mass On and smooth surface Sn experiences a rigid-body migration (n) with prescribed translational velocity U (here, the velocity of its attached point On) and angular velocity Ω(n). The resulting flow about the cluster has pressure field p and velocity field u with typical magnitude V > 0. For particles with length scale a the Reynolds number is Re = ρV a/µ. Assuming henceforth that Re = ρV a/µ  1 makes it possible to neglect all inertial effects and to consider that the flow (u, p) in the liquid domain D is governed by the steady Stokes problem

µ∇2u = ∇p and ∇.u = 0 in D, (1) (u, p) → (0, 0) far from the cluster. (2)

Of course, (1)-(2) must be supplemented with relevant boundary conditions on each particle surface. In practice, these conditions depend upon the nature of the fluid and/or of those surfaces. For instance, for a rarefied gas in the continuum regime or a liquid near a solid hydrophobic or lyophobic surface one allows the fluid to flow over the surface which is then called a slip surface. Here, we consider slip particles and adopt the following widely-employed and so-called Navier slip conditions

(n) (n) u(M) = U + Ω ∧ OnM + λn{ σ.n − (n. σ.n)n}/µ on each Sn (3) where n is the unit normal on Sn directed into the liquid and λn ≥ 0 the slip length of the surface Sn. This work will successively examine for the well-posed problem (1)-(3) the following issues: (i) Derive regularized boundary-integral equations for the unknown surface traction f = σ.n exerted on the cluster’s surface by the flow (u, p). (ii) Implement a boundary element technique to efficiently and accurately invert the encountered boundary-integral equations and calculate the net hydrodynamic force and torque exerted on each migrating slip particle. (iii) Test the advocated procedure against both analytical results (single slip sphere) and numerical results (single slip spheroid and a 2-sphere cluster) previously obtained in the literature using quite different methods. (iv) Present and discuss new numerical results for several clusters made of slip and/or non-slip particles.

4-15 4: Mini-Symposium: Analysis and applications of boundary element methods

BOUNDARY ELEMENT METHODS FOR ACOUSTIC RESONANCE PROBLEMS Gerhard Unger

Institute of Computational Mathematics, Graz University of Technology, Graz, Austria. [email protected]

We characterize acoustic resonances as eigenvalues of boundary integral operator eigen- value problems and use boundary element methods for their numerical approximation. Eigenvalue problem formulations for resonance problems which are based on standard boundary integral equations exhibit additional eigenvalues which are not resonances but eigenvalues of a related ”interior” eigenvalue problem. In practical computations it is for some typical applications hard to extract the resonances when using standard boundary integral formulations. In this talk we present regularized combined bound- ary integral formulations which only exhibit resonances as eigenvalues. We provide a comprehensive numerical analysis of the boundary element approximations of these eigenvalue problem formulations where general results of the discretization of eigenvalue problems for holomorphic Fredholm operator-valued functions are used [O. Steinbach, G. Unger: Convergence analysis of a Galerkin boundary element method for the Dirich- let Laplacian eigenvalue problem. SIAM J. Numer. Anal.,50 (2012), 710–728]. Finally we present some numerical examples.

4-16 4: Mini-Symposium: Analysis and applications of boundary element methods

BEM BASED SHAPE OPTIMIZATION USING SHAPE CALCULUS AND MULTIRESOLUTION ANALYSIS Jan Zapletal1, Kosala Bandara2a, Fehmi Cirak2b, G¨unther Of3c and Olaf Steinbach3d

1Department of Applied Mathematics, VSB – Technical University of Ostrava, Czech Republic, [email protected], 2Department of Engineering, University of Cambridge, [email protected], [email protected], 3Institute of Computational Mathematics, Graz University of Technology, [email protected], [email protected]

In the talk we present shape optimization based on the gradient information obtained by the shape calculus in the continuous setting. Since the shape of an object is given by its boundary and the shape gradient of the cost functional presented only involves evaluation of the Neumann data of the state and its adjoint, the boundary element method provides a suitable scheme for solving the associated boundary value problems. Together with the shape calculus we use the multiresolution analysis and perform the optimization on different levels of the corresponding surface mesh. This approach serves as a globalization strategy and in connection with mesh smoothening prevents ending up with badly shaped solutions.

4-17 5: Mini-Symposium: Boundary-Domain Integral Equations

5 Mini-Symposium: Boundary-Domain Integral Equa- tions

Organiser: Sergey Mikhailov

5-1 5: Mini-Symposium: Boundary-Domain Integral Equations

LOCALIZED BOUNDARY-DOMAIN INTEGRAL EQUATIONS APPROACH FOR DIRICHLET AND ROBIN PROBLEMS OF THE THEORY OF PIEZO-ELASTICITY FOR INHOMOGENEOUS SOLIDS Otar Chkadua

Mathematical Institute of I.Javakhishvili Tbilisi State University, 2 University str.,Tbilisi, Georgia and Sokhumi State University, 9 A.Politkovskaia str.,Tbilisi, Georgia. [email protected]

We consider the three–dimensional Dirichlet and Robin boundary-value problems (BVPs) of piezo-elasticity for anisotropic inhomogeneous solids and develop the gen- eralized potential method based on the localized parametrix method. Using Green’s representation formula and properties of the localized layer and volume potentials we reduce the Dirichlet and Robin BVPs to the localized boundary-domain integral equa- tions (LBDIE) systems. First we establish the equivalence between the original boundary value problems and the corresponding LBDIE systems. Afterwards, we establish that the localized boundary-domain integral operators obtained belong to the Boutet de Monvel alge- bra and with the help of the Vishik-Eskin theory, based on the factorization method (Wiener-Hopf method), we investigate corresponding Fredholm properties and prove invertibility of the localized operators in appropriate function spaces. This is a joint work with Sergey Mikhailov (Brunel University of London, UK) and David Natroshvili (Tbilisi Technical University, Georgia). Acknowledgements: This research was supported by grant No. EP/H020497/1: “Mathematical analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems” from the EPSRC, UK.

5-2 5: Mini-Symposium: Boundary-Domain Integral Equations

NUMERICS AND SPECTRAL PROPERTIES OF BOUNDARY DOMAIN INTEGRAL AND INTEGRO-DIFFERENTIAL OPERATORS IN 3D Richards Grzhibovskis1 and Sergey E. Mikhailov2

1 Department of Mathematics, Saarland University, Germany [email protected] 2 Department of Mathematical Sciences, Brunel University London, UK [email protected]

An elliptic boundary value problem with variable coefficients, where no fundamental solution is explicitly available, can be still reduced to a system of Boundary-Domain Integral or Integro-Differential Equations, BDI(D)Es, based on a parametrix (Levi function), cf. [3, 1, 4]. In this study, developing results of [2], we consider collocation discretization of BDI(D)E systems for the ”stationary difusion” problems with variable scalar-valued coefficient. In contrast to boundary integral formulation, a volume discretization is necessary even when the right hand side is zero, and the discretised layer potentials, volume potential and the remainder potential operators produce fully populated matri- ces. Two ways of avoiding prohibitively expensive second order complexity and storage requirements for the fully populated matrices are discussed. The first one is based on hierarchical matrix technique in conjunction with the adaptive cross approximation (ACA) procedure. The second way is related to a localized parametrix, which leads to localised BDI(D)Es that are reduced by discretisation to sparse matrices. We com- ment on the implementation details and report the results of numerical experiments in three-dimensional domains. Some of the considered non-localised BDI(D)E systems are of the second kind and solved by fast converging Neumann iterations, which is related with favourable spectral properties of the operators. We report the spectral properties of the corresponding discrete systems and compare them with the theoretical estimates. Note that the numerical spectral properties of some BDI(D)Es in 2D domains were presented in [5].

References

[1] O. Chkadua, S. E. Mikhailov, and D. Natroshvili. Analysis of direct boundary- domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. Journal of Integral Equations and Applications, 21(4):499–543, 2009.

[2] R. Grzhibovskis, S. Mikhailov, and S. Rjasanow. Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D. Computational Mechanics, Nr. 51, DOI: 10.1007/s00466-012-0777-8, pages 495–503, 2013.

5-3 5: Mini-Symposium: Boundary-Domain Integral Equations

[3] S. E. Mikhailov. Localized boundary-domain integral formulations for problems with variable coefficients. Engineering Analysis with Boundary Elements, 26:681– 690, 2002.

[4] S. E. Mikhailov. Analysis of united boundary-domain integro-differential and inte- gral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences, 29:715–739, 2006.

[5] S. E. Mikhailov and N. A. Mohamed. Numerical solution and spectrum of boundary- domain integral equation for the Neumann BVP with variable coefficient. Interna- tional Journal of Computer Mathematics, 89(11):1488–1503, 2012.

SPECTRAL PROPERTIES AND PERTURBATIONS OF BOUNDARY-DOMAIN INTEGRAL EQUATIONS Sergey E. Mikhailov

Department of Mathematics, Brunel University, London, UK [email protected]

The Dirichlet and Neumann BVPs for a variable-coefficient PDE on three-dimensional interior and exterior domains with compact boundaries are reduced to some direct Boundary-Domain Integro-Differential Equations (BDIDEs) of the second kind. Then the obtained BDIDEs are analysed in the weighted Sobolev (Beppo Levi type) function spaces more suitable for exterior domains and coinciding with the standard Sobolev spaces in interior domains. Equivalence of the BDIDEs to the original BVPs and the invertibility of the BDIDE operators are analysed. When the operators are not not invertible, they are still Fredholm with zero index, and we perturb them with some finite-dimensional operators, making the perturbed operators invertible and delivering a solution to of the original BVP. The spectral properties of the BDIDEs are then studied and some explicit conditions on the coefficient behaviour are given, insuring that the BDIE operator spectrum belongs to the unit circle and the Neumann iterations can be used to solve the corresponding BDIE systems. The analysis employs the methods developed in [1] - [6].

References

[1] Mikhailov, S. E., Finite-dimensional perturbations of linear operators and some applica- tions to boundary integral equations. Engineering Analysis with Boundary Elements, 23, 805–813 (1999).

[2] Mikhailov, S. E., Localized boundary-domain integral formulations for problems with variable coefficients. Eng. Analysis with Boundary Elements, 26, 681–690 (2002).

[3] Chkadua, O., Mikhailov, S. E., Natroshvili, D., Analysis of direct boundary-domain inte- gral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. Journal of Integral Equations and Applications, 21(4), 499–543 (2009).

5-4 5: Mini-Symposium: Boundary-Domain Integral Equations

[4] Mikhailov, S. E., Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences, 29, 715–739 (2006).

[5] Chkadua, O., Mikhailov, S. E., Natroshvili, D., Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks. Numerical Methods for Partial Differential Equations, 27(1), 121–140 (2011).

[6] Chkadua, O., Mikhailov, S. E., Natroshvili, D., Analysis of direct segregated boundary- domain integral equations for variable-coefficient mixed BVPs in exterior domains. Anal- ysis and Applications, 11(4), 2013 (to appear).

5-5 5: Mini-Symposium: Boundary-Domain Integral Equations

ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLE: BOUNDARY-DOMAIN INTEGRAL EQUATION APPROACH David Natroshvili

Georgian Technical University, Department of Mathematics, Tbilisi, Georgia [email protected]

We consider the time-harmonic acoustic wave scattering by a bounded layered anisotro- pic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. The material parameters and the refractive index are assumed to be discontinuous across the interfaces between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problems are formulated as boundary-transmission prob- lems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. We show that the boundary-transmission prob- lems with the help of localized potentials can be reformulated as a localized boundary- domain integral equations (LBDIE) systems and prove that the corresponding localized boundary-domain integral operators (LBDIO) are invertible. First we establish the equivalence between the original boundary-transmission prob- lems and the corresponding LBDIE systems which plays a crucial role in our anal- ysis. Afterwards, we establish that the localized boundary domain integral opera- tors obtained belong to the Boutet de Monvel algebra of pseudo-differential operators. And finally, applying the Vishik-Eskin theory based on the factorization method (the Wiener-Hopf method) we investigate Fredholm properties of the LBDIOs and prove their invertibility in appropriate function spaces. This invertibility property implies then existence and uniqueness results for the LBDIE systems and the corresponding original boundary-transmission problems. Beside a pure mathematical interest these results can be applied in constructing and analysis of numerical methods for solution of the LBDIE systems and thus the scattering problems in inhomogeneous anisotropic media.

This is a joint work with Sergey Mikhailov (Brunel University of London, UK) and Otar Chkadua (Tbilisi State University, Georgia).

Acknowledgements: This research was supported by grant No. EP/H020497/1: “Mathematical analysis of Localized Boundary-Domain Integral Equations for Variable- Coefficient Boundary Value Problems” from the EPSRC, UK.

5-6 6: Mini-Symposium: Computational Micromagnetics

6 Mini-Symposium: Computational Micromagnet- ics

Organisers: Dirk Praetorius and Thomas Schrefl

6-1 6: Mini-Symposium: Computational Micromagnetics

MAGNUM.FE: A MICROMAGNETIC FINITE-ELEMENT CODE BASED ON FENICS Claas Abert

Institut f¨urAngewandte Physik und Zentrum f¨urMikrostrukturforschung, Universit¨atHamburg, Jungiusstr. 11, D-20355 Hamburg, Germany [email protected]

Micromagnetism is the theory of choice for the description of magnetization dynamics on the nanometer scale. In contrast to domain theory this semiclassical continuum theory is able to resolve the structure of domain walls and describe switching pro- cesses. Also it is well suited for the treatment with numerical algorithms as opposed to quantum mechanical ab initio calculations. In this talk we present the complete micromagnetic finite-element code magnum.fe. magnum.fe is developed using the recently released general-purpose finite element pack- age FEniCS. Due to the high level of abstraction, the code is both very readable and extendable. Hence it delivers a good starting point for the implementation and testing of novel finite-element algorithms. Along with an overview over the software, we present a fully implicit and linear scheme for the integration of the Landau-Lifshitz-Gilbert equation, that is implemented in magnum.fe.

6-2 6: Mini-Symposium: Computational Micromagnetics

MULTISCALE SIMULATION OF MAGNETIC NANOSTRUCTURES Florian Bruckner

Institute of Solid State Physics, Vienna University of Technology, Vienna, Austria. [email protected]

Due to the ongoing miniaturization of modern magnetic devices, like GMR sensors, magnetic write heads, spintronic devices and so on, micromagnetic simulations gain more and more importance, since they are an essential tool to understand the behavior of magnetic materials in the nanometer scale. Using numerical simulations allows to optimize the micro-structure of such devices or to test new concepts prior to performing expensive experimental tests. A method is presented which allows to extend the micromagnetic model by ad- ditional macroscopic parts which are described in an averaged sense. In contrast to the microscopic parts which are described by Landau-Lifshitz-Gilbert (LLG) equa- tions, these macroscopic parts are based on classical magnetostatic Maxwell equations, which could be extended to a full Maxwell description in a straight-forward way. The averaged description using Maxwell equations allows to overcome the upper bound for the discrete element sizes, which is intrinsic to the micromagnetic models, since the detailed domain structure of the ferromagnetic material needs to be resolved. Combining microscopic and macroscopic models and solving the corresponding equations simultaneously provides a multiscale method, which allows to handle prob- lems of a dimension, which would otherwise be far out of reach. A basic prerequisite for the application of the method, is that microscopic and macroscopic parts can be separated into disjoint regions. The performance of the implemented algorithm is demonstrated by the simulation of the transfer curve of a magnetic recording read head, as it is built into current hard drives. Independent of the former approach the use of parallelized algorithms allow the handle larger problems. A shared-memory-parallelization of hierarchical matrices is demonstrated, since these require a large amount of the storage consumption as well as of the computation time for typical simulations. For the setup of the matrices a nearly perfect parallelization could be reached, whereas for the matrix-vector-multiplication the computation time stagnates at a few computation cores, due to the restricted main memory bandwidth.

References

[1] F. Bruckner, C. Vogler, M. Feischl, D. Praetorius, B. Bergmair, T. Huber, M. Fuger, D. S¨uss,“3D FEM-BEM-coupling method to solve magnetostatic Maxwell equations”, J. Magn. Magn. Mater., 324 (2012), 1862-1866.

[2] F. Bruckner, M. Feischl, T. F¨uhrer, P. Goldenits, M. Page, D. Praetorius, D. Suess, 2012), “Multiscale modeling in micromagnetics: well-posedness and numerical in- tegration”, arXiv:1209.5548 [math.NA].

6-3 6: Mini-Symposium: Computational Micromagnetics

[3] F. Bruckner, C. Vogler, B. Bergmair, T. Huber, M. Fuger, D. S¨uss,M. Feischl, T. F¨uhrer,M. Page, D. Praetorius, “Combining micromagnetism and magnetostatic Maxwell equations for multiscale magnetic simulations”, commited to J. Magn. Magn. Mater., (2013).

FINITE ELEMENT AND BOUNDARY ELEMENT METHOD IN MAGNETIC SPIN TRANSPORT AND MAGNETIC HYBRID STRUCTURES Gino Hrkac1, Marcus Page2 and Dieter Suess2

1College of Engineering, Mathematics and Physical Sciences, University of Exeter, Devon, UK [email protected] 2Vienna University of Technology, Austria

In this paper we investigate the difference in the Slonzewski spin torque approach and the change of spin accumulation based on a diffusion equation derived by Zhang, Levy and Fert and at an interface of a magnetic to a non-magnetic material, in a micromag- netic framework, by considering two cases; first a spin-accumulation due to current and second due to field excitation; refereed to as spin-pumping. In spin polarized transport models, the magnetization was assumed to be constant / pinned in one of the layers and the polarization was introduced as an extra field that is part of the effective field. Spin diffusion and interfacial effects were neglected but it was shown that these effects are important in magnetoresistance experiments. Zhang et al. introduced a model for the relaxation of a coupled spin-magnetization system, where they considered the one-dimensional case. We start from a spin dynamics equation that is derived from a Boltzman transport equation and couple it to the Land Lifshitz Gilbert equation. We do a comparable study between the diffusion approach and the Slonzewski approach that is derived from circuit theory. In the first case of our study we assume a constant electric current and mapped the spin-accumulation profiles over the interface and for the second case we omitted the electric current and assume only an applied field and a time varying magnetization. This leads to a reduced expression of the spin accumulation current in the diffusion equation, meaning that the spin accumulation is not fully absorbed in the ferromagnet and unabsorbed transverse spin accumulation diffuses over the interface into the metal layer and leads to a modified spin-torque. Also we discuss the effect of symmetric and asymmetric spin torque effects on the vortex oscillations in a spin-valve system in the presence of two vortices and its effect on the frequency and linewidth. We will show that the mutal interaction between two vortices, considering a non-linear current distribution, pseudo-diffusion model, can lead to non-linear oscillation regimes.

The authors gratefully acknowledge financial support from the Royal Society UK and the WWTF.

6-4 6: Mini-Symposium: Computational Micromagnetics

COUPLING AND NUMERICAL INTEGRATION OF LLG Marcus Pagea and Dirk Praetoriusb

Vienna University of Technology, Institute for Analysis and Scientific Computing, Vienna, Austria [email protected], [email protected]

In our talk, we give an overview on our recent preprints [arXiv:1209.5548], [arXiv: 1303.4009], and [arXiv:1303.4060]. These are concerned with the numerical integration of extended formulations of the Landau-Lifshitz-Gilbert equation (LLG), where the coupling of LLG with other PDEs is analyzed. We extend a recent algorithm of [Alouges 2008] who considered the exchange-only formulation of LLG and introduced and analyzed an implicit numerical integrator which only requires the solution of one linear system per time step. Moreover, the proposed integrator is unconditionally convergent, i.e. no coupling of spatial mesh-size h and time step-size k are imposed for stability. The convergence proof is constructive in the sense that it also proves existence of weak solutions. Independently, [Goldenits-Praetorius-Suess 2011] and [Alouges-Kritsikis-Toussaint 2012] included the explicit integration of uniaxial anisotropy, demagnetization, and applied external fields (i.e. the effective field is extended, yet linear) into the overall Alouges-type integrator. As before, they proved unconditional convergence. In the above preprints, we further extend the Alouges integrator to show its full po- tential. By exploiting an abstract framework, we can cover general field contributions that might be nonlinear, non-local, and/or time-dependent. Applications include mul- tiscale modelling with LLG, coupling of LLG to the full Maxwell system or some eddy- current regime, or even to the conservation of momentum equation to include magne- tostrictive effects. Moreover, for the coupling of LLG with time-dependent PDEs, one focus is on the decoupling of the respective time integration. For the Maxwell-LLG system, for instance, only two linear systems have to be solved per time step, one for the LLG part and one for the Maxwell part. Even in this general setting, we still prove unconditional convergence, and our proof also provides the existence of weak solutions.

6-5 6: Mini-Symposium: Computational Micromagnetics

A NONLOCAL PARABOLIC AND HYPERBOLIC MODEL FOR TYPE-I SUPERCONDUCTORS Karel Van Bockstala and Marian Slodiˇckab

Research Group NaM2, Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Ghent, Belgium [email protected], [email protected]

A vectorial nonlocal linear parabolic and hyperbolic problem with applications in su- perconductors of type-I is studied. A superconductive material of type-I occupies a bounded domain Ω ⊂ R3 with a Lipschitz continuous boundary ∂Ω. The full Maxwell’s equations (δ = 1) and quasi-static Maxwell’s equations (δ = 0) for linear materials are considered. They can be written as

∇ × H = J + δ∂tD = J + δε∂tE; (1)

∇ × E = −∂tB = −µ∂tH.

The current density J is supposed to be the sum of a normal and a superconducting part, that is J = J n + J s. The normal density current J n is required to satisfy Ohm’s law J n = σE. The nonlocal representation of the superconductive current J s by Eringen [1] is considered. This representation identifies the state of the superconductor, at time t, with the magnetic field H(·, t) and is given by the linear functional Z 0 0 0 0 J s(x, t) = σ0 (|x − x |)(x−x )×H(x , t)dx =: −(K0?H)(x, t), (x, t) ∈ Ω×(0,T ). Ω Taking the curl of (1) results into the following parabolic (δ = 0) and hyperbolic (δ = 1) integro-differential equation

δεµ∂ttH + σµ∂tH + ∇ × ∇ × H + ∇ × (K0 ? H) = 0. (2)

A new convolution kernel is derived, namely ∇ × J s = −K ? H when H is divergence free. The positive definiteness of the kernel K is shown. Taking into account the previous consideration, equation (2) can be rewritten as

δεµ∂ttH + σµ∂tH − ∆H + K ? H = 0.

The well-posedness of both problems is discussed under low regularity assumptions and the error estimates for various time-discrete schemes (based on backward Euler approximation) are established.

References

[1] A.C. Eringen. Electrodynamics of memory-dependent nonlocal elastic continua. J. Math. Phys., 25:3235–3249, 1984.

6-6 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

7 Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

Organisers: Paola Antonietti, Paul Houston and Ilaria Perugia

7-1 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

ENERGY STABILITY FOR DISCONTINUOUS GALERKIN APPROXIMATION OF A PROBLEM IN ELASOTODYNAMICS Paola F. Antonietti1, Blanca Ayuso de Dios2, Ilario Mazzieri3 and Alfio Quarteroni4,5

1 MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy [email protected] 2 Centre de Recerca Matem´atica,UAB Science Faculty, 08193 Bellaterra, Barcelona, Spain [email protected] 3 MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy [email protected] 4 MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 5 CMCS, Ecole Polytechnique Federale de Lausanne (EPFL), Station 8, 1015 Lausanne, Switzerland [email protected]

We introduce a family of semidiscrete-DG methods for the approximation of a general elastodynamics problem. We discuss the energy stability and present the a-priori error analysis of the methods. We also propose some schemes that preserve the total energy of the system and present some numerical experiments that verify the theory.

7-2 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

STAGGERED DISCONTINUOUS GALERKIN METHODS FOR MAXWELL’S EQUATIONS Eric Chung

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR. [email protected]

In this talk, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations is presented. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non- staggered method is carried out and shows that our method has better accuracy and efficiency.

7-3 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

EFFICIENT DISCONTINUOUS GALERKIN METHOD FOR METEOROLOGICAL APPLICATIONS Andreas Dedner1 and Robert Kl¨ofkorn2

1Mathematics Institute and Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, UK [email protected], 2Computational Math Group, UCAR, Boulder, USA [email protected]

In this talk we will introduce a dynamic core for local weather predication based on the Discontinuous Galerkin method. The method is implemented within the Dune software framework (www.dune-project.org). It allows the simulation of the compressible multi species Navier Stokes equations on general 3D meshes in parallel. Special mechanisms are included to allow the simulation of advection dominated flow and for including local grid adaptation. We will demonstrate that the code allows for efficient, highly scalable parallel simulations. To achieve this goal we have developed a stabilization mechanism for the advection dominated case which works on general unstructured meshes. For the diffusion operator we have developed a highly efficient discretization approach. To prove the effectiveness and efficiency of our Dune base numerical core, we compare it with the dynamical core of the COSMO local area model. COSMO is an operational code, used by many weather services. This comparison was performed in cooperation with the German Weather Service. We compare the performance of the very different cores based on a number of standard meteorological benchmarks. We will demonstrate that the Dune core shows a higher numerical convergence rate. The high subscale resolution of the DG method means that Dune produces a lower error when fixing the grid resolution or the number of degrees of freedom.

7-4 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

DISCONTINUOUS GALERKIN METHODS FOR PHASE FIELD MODELS OF MOVING INTERFACE PROBLEMS Xiaobing Feng

Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, U.S.A. [email protected]

This talk is concerned with some new convergence results for interior penalty discon- tinuous Galerkin (IPDG) approximations of two types of phase field models which are described respectively by the 2nd order Allen-Cahn equations and the 4th order Cahn- Hilliard equations. The main result to be discussed is the convergence of the numerical interfaces to the sharp interfaces of the limit models of moving interface problems (namely, the mean curvature flow and the Hele-Shaw flow) as both the numerical mesh parameters and the phase field parameter (called the interaction length) tend to zero. The crux for establishing the result is to derive, by a nonstandard technique, error estimates for the IPDG solutions which blows up only polynomially (instead of expo- nentially) in the reciprocal of the phase field parameter. This is a jointly work with Yukun Li of the University of Tennessee at Knoxville.

DISCONTINUOUS GALERKIN METHODS FOR NON-LINEAR INTERFACE PROBLEMS Emmanuil H. Georgoulis

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom [email protected]

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. The case of fast reactions is also included. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment with only local Lipschitz conditions on the nonlinear reaction terms, equipped with respective initial and boundary conditions, is considered. General nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method for this problem is analysed both in the space-discrete and in fully discrete settings for the case of, possibly, fast reactions. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds. The talk is based on joint work with Andrea Cangiani (Leicester) and Max Jensen (Durham).

7-5 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

ON THE CONVERGENCE OF ADAPTIVE DISCONTINUOUS GALERKIN METHODS

Thirupathi Gudi1 and Johnny Guzm´an2

1Department of Mathematics, Indian Institute of Science Bangalore, 560012 India [email protected] 2Division of Applied Mathematics, Brown University, Providence RI 02912, USA

Establishing the convergence and optimality of adaptive finite element methods has been an important and active research topic for more than a decade. In the study, one of the key step is to establish a contraction property of the form:

kehkh + βηh ≤ ρ (keH kH + βηH ) , where eh and ηh (or eH and ηH ) are the exact error and the error estimator, respectively, on the triangulation Th (or TH ). The contraction property is proved for the standard finite element method, for the nonconforming finite element method and for the mixed finite element methods in the literature. Also a contraction property is proved for the symmetric interior penalty method in [Karakashian and Pascal, SINUM, 45 (2007), pp. 641–665], [Hoppe, Kanschat and Warburton, SINUM, 47 (2009), pp. 534–550] and [Bonito and Nochetto, SINUM, 48 (2010), pp. 734–771]. The common issue with these three articles is that the contraction property is derived assuming the penalty parameters are sufficiently large (i.e. larger than what is needed for stability of the method). In this talk, we present contraction properties for various symmetric weakly penalized discontinuous Galerkin methods only assuming that the penalty parameters are large enough to guarantee stability of the method. For example, in the case of the LDG method the stabilizing parameters only have to be positive. This is achieved by a new marking strategy that uses an auxiliary solution obtained by post-processing the discontinuous Galerkin solution.

7-6 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

A COCHAIN COMPLEX FOR INTERIOR PENALTY METHODS: ERROR ESTIMATES AND MULTIGRID THROUGH DIFFERENTIAL RELATIONS Guido Kanschata and Natasha Sharmab

IWR, Ruprecht-Karls-Universit¨atHeidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany. [email protected], [email protected]

We show that the recently developed divergence-conforming methods for the Stokes problems have discrete stream functions. These stream functions in turn solve a contin- uous interior penalty problem for biharmonic equations. The equivalence is established for the most common methods in two dimensions based on interior penalty terms. We show that this relation can be exploited to transfer results on multigrid methods and on error estimates between the two schemes. Through the numerical results, we will illustrate the efficiency of the methods obtained from these relations.

GENERALIZED DG-METHODS FOR HIGHLY INDEFINITE HELMHOLTZ PROBLEMS Jens Markus Melenk1, Asieh Parsania2 and Stefan A. Sauter3

1Institute f¨urAnalysis und Scientific Computing, Technische Universit¨atWien, Wien, Austria. [email protected] 2Institut f¨urMathematik, Universit¨atZ¨urich, Z¨urich, Switzerland. [email protected] 3Institut f¨urMathematik, Universit¨atZ¨urich, Z¨urich, Switzerland. [email protected]

We develop a stability and convergence theory for the DG-formulation of a highly indefinite Helmholtz problem. The theory covers conforming as well as nonconform- ing generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, we prove that the DG-formulation admits a unique solution under much weaker con- ditions. As an application we present the error analysis for the hp-version of the finite element method explicitly in terms of the mesh width h, polynomial degree p and wave number k. It is shown that the optimal convergence order estimate is obtained under √ the conditions that kh/ p is sufficiently small and the polynomial degree p is at least O(log k).

7-7 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

CONVERGENCE OF HIGH ORDER METHODS FOR THE MISCIBLE DISPLACEMENT PROBLEM Beatrice Riviere

Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, Texas, 77005, USA. [email protected]

Miscible displacement flow is one important part of enhanced oil recovery. A poly- meric solvent is injected in the reservoir and it mixes with the trapped oil. Accurate simulation of the displacement of the fluid mixture in heterogeneous media is needed to optimize oil production. The miscible displacement is mathematically modeled by a pressure equation (ellip- tic) and a concentration equation (parabolic) that are coupled in a nonlinear fashion. The convergence analysis of numerical methods applied to the miscibledisplacement is challenging because the diffusion-dispersion coefficient in the concentration equation is unbounded. In this work, we formulate and analyze discontinuous Galerkin in time methods coupled with several finite element methods (including mixed finite elements and dis- continuous Galerkin) for the miscible displacement. The diffusion-dispersion coefficient is not assumed to be bounded. Other coefficients in the problem are not assumed to be smooth. Convergence of the numerical solution is obtained using a generalization of the Aubin-Lions compactness theorem. The Aubin-Lions theorem is not applicable since the numerical approximations are discontinuous in time. Numerical examples with varying order in space and time are also given.

7-8 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

HP-MULTIGRID AS SMOOTHER ALGORITHM FOR HIGHER ORDER DISCONTINUOUS GALERKIN DISCRETIZATIONS OF ADVECTION-DOMINATED FLOWS Jaap van der Vegt1 and Sander Rhebergen2

1University of Twente, Department of Applied Mathematics 7500 AE, Enschede, The Netherlands [email protected] 2University of Oxford, Mathematical Institute 24-29 St Giles’, Oxford, OX1 3LB, UK [email protected]

Higher order accurate space-time discontinuous Galerkin discretizations are well suited for free boundary problems since they remain conservative on moving and deforming meshes and are well suited for solution adaptive and parallel computations. The space- time DG discretization results, however, in an implicit discretization in time, which requires the solution of a (non)linear algebraic system. In this presentation the recently developed hp-Multigrid as Smoother (hp-MGS) algorithm for higher order accurate DG discretizations of advection-dominated flows will be discussed. The main feature of this algorithm is that it uses semi-coarsening h-multigrid as smoother for p-multigrid. The development of the hp-MGS algorithm strongly relies on a detailed multilevel Fourier analysis of the full multigrid algorithm, which provides the essential information to optimize the coefficients in the semi-implicit Runge-Kutta smoother. In addition, this multilevel analysis gives detailed information on the spectrum and operator norms of the error transformation operator. In this presentation we will consider both hexahedral and prismatic space-time elements. The newly developed multigrid algorithm will be demonstrated on various problems with thin boundary layers that require significantly stretched meshes.

References

[1] J.J.W. van der Vegt and S. Rhebergen, HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis, Journal of Computational Physics, Vol. 231, pp. 7537- 7563, 2012.

[2] J.J.W. van der Vegt and S. Rhebergen, HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother, Journal of Computational Physics, Vol. 231, pp. 7564-7583, 2012.

7-9 7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

MIXED HP -DGFEM FOR LINEAR ELASTICITY IN 3D Thomas P. Wihlera and Marcel Wirzb

Mathematics Institute, University of Bern, Switzerland. [email protected], [email protected]

We consider mixed hp-discontinuous Galerkin FEM for linear elasticity in axiparallel polyhedral domains in R3. In order to resolve possible corner, edge, and corner-edge sin- gularities, anisotropic geometric edge meshes consisting of hexahedral elements are ap- plied. We discuss inf-sup stability results on both the continuous as well as the discrete level. In addition, under certain realistic assumptions (for analytic data) on the regular- ity of the exact solution and based on an hp-interpolation analysis from [D. Sch¨otzau, C. Schwab, and TPW, hp-dGFEM for second-order elliptic problems in polyhedra II: Exponential convergence, accepted for publication in SINUM], we prove that the proposed DG schemes converge at an exponential rate in terms of the fifth root of the number of degrees of freedom [TPW and MW, Mixed hp-discontinuous Galerkin FEM for linear elasticity and Stokes flow in three dimensions, Math. Models Methods Appl. Sci. 22 (2012), no. 8]. A number of numerical experiments will illustrate the theory.

7-10 8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

8 Mini-Symposium: Discontinuous Galerkin meth- ods in fluid flows

Organisers: Aycil Cesmelioglu and Sander Rhebergen

8-1 8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

HYBRIDIZABLE DISCONTINUOUS GALERKIN METHODS FOR THE INCOMPRESSIBLE OSEEN AND NAVIER-STOKES EQUATIONS Aycil Cesmelioglu1, Bernardo Cockburn2, Ngoc Cuong Nguyen3a and Jaime Peraire3b

1Department of Mathematics and Statistics, Oakland University, Rochester, MI, US [email protected] 2School of Mathematics, University of Minnesota, Minneapolis, MN, US [email protected] 3Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, US [email protected], [email protected]

The aim of this work is to analyze hybridizable discontinuous Galerkin (HDG) methods for the incompressible stationary Navier-Stokes problem. First, we study HDG methods for the Oseen equations. We show optimal convergence for the velocity, its gradient and the pressure by using same degree polynomial approximations for all the unknowns. Furthermore, after a postprocessing, we obtain a H(div)-conforming, divergence-free velocity which converges with an additional order. We show numerical examples to validate the theoretical convergence rates. Finally, we discuss the extension of these results to the Navier-Stokes case using a sequence of Oseen approximations.

COMMUTING DIAGRAMS FOR THE TNT ELEMENTS ON CUBES Bernardo Cockburn1 and Weifeng Qiu2

1School of Mathematics, University of Minnesota 2Department of Mathematics, City University of Hong Kong [email protected]

We present commuting diagrams for the de Rham complex for new elements defined on cubes which use tensor product spaces. The distinctive feature of these elements is that, in sharp contrast with previously known results, they have the TiNiest spaces containing Tensor product spaces of polynomials of degree k, hence their acronym TNT. In fact, the local spaces of the TNT elements differ from the standard tensor product spaces by spaces whose dimension is a small number independent of the degree k. Such number is 7 (number of vertices of the cube minus one) for the space associated with the divergence operator, 18 (number of faces of the cube times the number of vertices of a face minus one) for the space associated with the curl operator, and 12 (number of edges of the cube times the number of vertices of an edge minus one) for the space associated with the gradient operator.

8-2 8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

DEVELOPMENT AND VALIDATION OF A DISCONTINUOUS GALERKIN WAVE PREDICTION MODEL Ethan Kubatkoa and Angela Nappib

Department of Civil, Environmental and Geodetic Engineering, The Ohio State University, Columbus, OH, USA [email protected], [email protected]

In this talk, we present the development and validation of a discontinuous Galerkin (DG) method for a wave prediction model known as the GLERL–Donelan wave model. Originally formulated at the Canadian Centre for Inland Waters and the US National Oceanic and Atmospheric Administration’s Great Lakes Environmental Research Lab- oratory (GLERL), the GLERL–Donelan model is a relatively simple parametric wave model that has historically formed the basis of the US National Weather Service’s Great Lakes wave forecasts. In contrast to spectral and most other parametric wave models, which solve the so-called spectral action balance equation, the GLERL–Donelan wave model is based on the conservation of total wave momentum. This formulation avoids the need to solve the action balance equation over a (usually large) set of discrete frequency components, which can make spectral-based models prohibitively expensive from a computational perspective. The development and numerical implementation of a DG method for the solution of the GLERL–Donelan wave model will be discussed, and model validation results will be presented and compared to a spectral-based, third- generation wave model in terms of accuracy and computational cost.

8-3 8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

COUPLING OF STOKES AND DARCY FLOWS USING DISCONTINUOUS GALERKIN AND MIMETIC FINITE DIFFERENCE METHOD Konstantin Lipnikov1, Danail Vassilev2 and Ivan Yotov3

1Applied Mathematics and Plasma Physics Group, Theoretical Division, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, [email protected] 2Mathematics Research Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter, EX4 4QF, UK [email protected] 3Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260, USA, [email protected]

We present a numerical method for coupling Stokes and Darcy flows based on discontin- uous Galerkin (DG) elements for Stokes and mimetic finite difference (MFD) methods for Darcy. Both methods are locally mass conservative and can handle irregular grids. The MFD methods are especially suited for flow in heterogeneous porous media, as they provide accurate approximation for both pressure and velocity and can handle discontinuous coefficients as well as degenerate and non-convex polygonal elements. We develop DG polygonal elements for Stokes, allowing for coupled discretizations on polygonal grids. Optimal convergence is obtained for the coupled numerical method and confirmed computationally.

8-4 8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

LOCAL DISCONTINUOUS GALERKIN METHOD FOR INKJET DROP FORMATION AND MOTION Tatyana Medvedevaa, Onno Bokhove and Jaap van der Vegt

1University of Twente, Department of Applied Mathematics 7500 AE, Enschede, The Netherlands [email protected]

In inkjet printing accurate control of droplet formation is crucial to obtain a high printing quality. In order to investigate this complex process the one-dimensional model proposed by Eggers and Dupont [1] was discretized using a Local Discontinuous Galerkin method. Viscosity and free surface terms are included in this discretization. Since the resulting discrete system is very stiff an implicit time integration method is used in combination with a Newton method. The newly developed LDG method is verified with detailed computations of the growth rate of the free surface and other exact solutions. Also, simulation of the droplet formation in inkjet will be presented.

References

[1] J. Eggers, T.F. Dupont, Drop formation in a one-dimensional approximation of the Navier-Stokes equation, J. Fluid Mech.262 (1994) 205-221.

SPACE-TIME (H)DG METHODS FOR INCOMPRESSIBLE FLOWS Sander Rhebergen1, Bernardo Cockburn2 and Jaap van der Vegt3

1Mathematics Institute, University of Oxford, UK. [email protected] 2School of Mathematics, University of Minnesota, USA. [email protected] 3Department of Applied Mathematics, University of Twente, Netherlands. [email protected]

In this talk I will discuss a new Discontinuous Galerkin (DG) method, namely the Hy- bridizable DG (HDG) method. We recently extended the HDG method to a space-time formulation allowing efficient and accurate computations on deforming grids/domains. I will introduce the method for the Incompressible Navier-Stokes (INS) equations. Re- sults and efficiency of the space-time HDG method for the INS equations on deforming domains will then be compared to those of the space-time DG method.

8-5 8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE PROPAGATION OF PHASE TRANSITION IN SOLIDS Lulu Tiana, Yan Xub, J.G.M. Kuertenc and Jaap van der Vegtd

Mathematics of Computational Science, Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. [email protected], [email protected], [email protected], [email protected]

In this presentation, we will discuss a local discontinuous Garlerkin (LDG) finite ele- ment method for the solution of a hyperbolic-elliptic system modeling the propagation of phase transition in solids. Viscosity and capillarity terms are added to select the physically relevant solution. The L2−stability of the LDG method is proven for ba- sis functions of arbitrary polynomial order. In addition, using a priori error analysis, it is proven that the LDG discretization converges at optimal order if the solution is sufficiently smooth. Also, results of a linear stability analysis to determine the time step are presented. To obtain a reference exact solution we solved a Riemann prob- lem for a trilinear strain-stress relation using a kinetic relation to select the unique admissible solution. This exact solution contains both shocks and phase transitions. The LDG method is demonstrated by computing several model problems representing phase transition in solids and in fluids with a Van der Waals equation of state. The results show the convergence properties of the LDG method.

8-6 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

9 Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computa- tion

Organisers: Stefano Giani, Luka Grubisic and Jeffrey Ovall

9-1 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

GUARANTEED LOWER BOUNDS FOR EIGENVALUES Carsten Carstensen1 and Joscha Gedicke2

1Institut f¨urMathematik, Humboldt-Universit¨atzu Berlin, Unter den Linden 6, 10099 Berlin, Germany and Department of Computational Science and Engineering, Yonsei University, 120–749 Seoul, Korea, [email protected] 2Institut f¨urMathematik, Humboldt-Universit¨atzu Berlin, Unter den Linden 6, 10099 Berlin, Germany, [email protected]

This talk introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the cor- responding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. Numerical examples demonstrate the reliability of the guaranteed error control even with inexact solve of the algebraic eigenvalue problem. This motivates an adaptive algorithm which moni- tors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error. The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiency indices as small as 1.4.

9-2 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

ADAPTIVE PATH-FOLLOWING METHOD FOR NONLINEAR PDE EIGENVALUE PROBLEMS Carsten Carstensen1, Joscha Gedicke2, V. Mehrmann3a and Agnieszka Miedlar3b

1Humboldt-Universit¨atzu Berlin, Unter den Linden 6, 10099 Berlin, Germany and Department of Computational Science and Engineering, Yonsei University, 120–749 Seoul, Korea. [email protected] 2Humboldt-Universit¨atzu Berlin, Unter den Linden 6, 10099 Berlin, Germany. [email protected] 3Technische Universit¨atBerlin, Institut f¨urMathematik, MA 4-5, Strasse des 17. Juni 136, 10623 Berlin, Germany. [email protected], [email protected]

In this talk we introduce a new approach combining the adaptive finite element method with the homotopy method to determine the eigenpairs for problems arrising in acous- tic field computations with proportional damping. The presented adaptive homotopy approach emphasizes the need of the multi-way adaptation based on different errors, i.e., the homotopy, the discretization and the iteration error. All our statements are illustrated with several numerical examples.

ADAPTIVE NONCONFORMING CROUZEIX-RAVIART FEM FOR EIGENVALUE PROBLEMS Carsten Carstensena, Dietmar Gallistlb and Mira Schedensack c

Department of Mathematics, Humboldt-Universit¨atzu Berlin, Germany. [email protected], [email protected], [email protected]

The nonconforming eigenvalue approximation is of high practical interest because it allows for guaranteed upper and lower eigenvalue bounds and for a convenient computa- tion via a consistent diagonal mass matrix in 2D. The first main result is a comparison which states equivalence of the error of the nonconforming eigenvalue approximation with its best-approximation error and its error in a conforming computation on the same mesh. The second main result is optimality of an adaptive algorithm for the ef- fective eigenvalue computation for the Laplace operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of a nonlinear ap- proximation class. The analysis includes an inexact algebraic eigenvalue computation on each level of the adaptive algorithm which requires an iterative algorithm and a controlled termination criterion.

9-3 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

COMPUTATION OF GROUND STATES OF SCHRODINGER¨ OPERATOR WITH LARGE MAGNETIC FIELDS Monique Dauge

IRMAR, Universit´ede Rennes 1, Campus de Beaulieu, Rennes, France [email protected]

The Schr¨odingeroperator with magnetic field takes the form

2 PA = (i∇ + A)

d where A is a vector field. This operator PA is set on a domain Ω of R (d = 2 or 3) and completed by natural boundary conditions (Neumann). Denote it by PΩ,A. The ground states of PΩ,A are the eigenpairs (λ, ψ)

2 (i∇ + A) ψ = λψ in Ω and (i∂n + n · A)ψ = 0 on ∂Ω associated with the lowest eigenvalues λ. If Ω is bounded, PΩ,A is positive self-adjoint with compact resolvent. If Ω is simply connected, its eigenvalues depend only on the magnetic field B defined as B = curl A. The eigenvectors corresponding to two different instances of A for the same B are deduced from each other by a gauge transform. The ground states of PΩ,A for large B are related to the solutions of the linearized equations of Ginzburg-Landau for the determination of critical fields for which super- conductivity arises. Introducing a (small) parameter h and setting

2 Ph,Ω,A = (ih∇ + A) with Neumann b.c. on ∂Ω, we get the relation 2 Ph,Ω,A = h PΩ,A/h linking the problem with large magnetic field to the semi-classical limit h → 0. If B is constant (and non-zero),√ the eigenvectors concentrate at the boundary as h → 0 with the length scale h in the normal direction. In dimension d = 2, the concentration takes place around points of maximal curvature. If Ω has convex cor- ners, they produce even stronger concentration of eigenvectors and, in general, the eigenvectors ψ = ψh have a two-scale structure of the form as h → 0

x − x0  iΦ(x)/h ψh(x) = Ψ √ e , x → x0 (x0 corner). h The rapidly oscillating phase Φ(x)/h makes it difficult to compute accurately the eigen- pairs as h → 0. We will show on examples that uniform p-version of finite elements perform well and is able to capture interesting intertwining behavior of eigenvalues in the presence of symmetries. This talk is based on references [1, 2] and work in progress [3].

9-4 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

References

[1] V. Bonnaillie-Noel¨ and M. Dauge, Asymptotics for the low-lying eigenstates of the Schr¨odingeroperator with magnetic field near corners, Ann. Henri Poincar´e, 7 (2006), pp. 899–931.

[2] V. Bonnaillie-Noel,¨ M. Dauge, D. Martin, and G. Vial, Computations of the first eigenpairs for the Schr¨odingeroperator with magnetic field, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 3841–3858.

[3] V. Bonnaillie Noel,¨ M. Dauge, and N. Popoff, Polyhedral bodies in large magnetic fields.

FINITE ELEMENT ANALYSIS OF A NON-SELF-ADJOINT QUADRATIC EIGENVALUE PROBLEM Christian Engstr¨om

Department of Mathematics and Mathematical Statistics, Ume˚aUniversity, Sweden [email protected]

In this talk we present Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the p-version and the h-version of the finite element method confirm the theoretical convergence rates

9-5 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

SOLVING AN ELLIPTIC EIGENVALUE PROBLEM VIA AUTOMATED MULTI-LEVEL SUB-STRUCTURING AND HIERARCHICAL MATRICES Peter Gerdsa and Lars Grasedyckb

Institut f¨urGeometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany. [email protected], [email protected]

To solve an elliptic eigenvalue problem we combine the automated multi-level sub- structuring (or short AMLS) method [1, 3, 4] with the concept of hierarchical matrices (or short H-matrices) [2]. AMLS is a sub-structuring method which projects the dis- cretized eigenvalue problem in a small subspace. A reduced eigenvalue problem has to be computed which delivers approximate solutions of the original problem. Several practical examples show that the AMLS method can be much faster than the commonly used shift-invert block Lanczos algorithm. Whereas the AMLS method is very effective in the two-dimensional case, the AMLS method is getting very expensive in the three-dimensional case, due to the fact that it computes the reduced eigenvalue via dense matrix operations. But here hierarchical matrices can help. H-matrices are a data-sparse approxima- tion of dense matrices which e.g. result from the discretisation of the inverse of elliptic partial differential operators. The main advantage of H-matrices is that they allow matrix algebra in almost linear complexity. In this talk we present how the AMLS method is combined with the H-matrices and how the reduced eigenvalue problem is computed by the fast H-matrix algebra. Beside the discretisation error two additional errors occur, the projection error of the AMLS method and the error caused by the H-matrix approximation. These errors are controlled by several parameters. The influence of these parameters will be investigated in examples. Furthermore we will show in examples that we can compute the reduced eigenvalue problem in the three-dimensional case in almost linear complexity.

References

[1] Jeffrey K. Bennighof and R. B. Lehoucq. An automated multilevel substructur- ing method for eigenspace computation in linear elastodynamics. SIAM J. Sci. Comput., 25(6):2084–2106 (electronic), 2004.

[2] Steffen B¨orm,Lars Grasedyck, and Wolfgang Hackbusch. Introduction to hierar- chical matrices with applications. Engineering Analysis with Boundary Elements, 27(5):405 – 422, 2003.

[3] Weiguo Gao, Xiaoye S. Li, Chao Yang, and Zhaojun Bai. An implementation and evaluation of the AMLS method for sparse eigenvalue problems. ACM Trans. Math. Software, 34(4):Art. 20, 28, 2008.

9-6 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

[4] Chao Yang, Weiguo Gao, Zhaojun Bai, Xiaoye S. Li, Lie-Quan Lee, Parry Hus- bands, and Esmond Ng. An algebraic substructuring method for large-scale eigen- value calculation. SIAM J. Sci. Comput., 27(3):873–892 (electronic), 2005.

AUXILIARY SUBSPACE ERROR ESTIMATION FOR HIGH-ORDER FINITE ELEMENT EIGENVALUE APPROXIMATIONS Stefano Giani1, Luka Grubiˇsi´c2, Harri Hakula3 and Jeffrey S Ovall4

1School of Engineering and Computing Sciences, University of Durham, UK [email protected] 2Department of Mathematics, University of Zagreb, Croatia [email protected] 3Department of Mathematics and Systems Analysis, Aalto University, Finland [email protected] 4Department of Mathematics, University of Kentucky, USA [email protected]

Given a bounded open set Ω ⊂ Rd (d = 2, 3), we consider high-order (p and hp) finite element approximations of eigenvalues and their corresponding invariant subspaces for positive and self-adjoint second-order elliptic operators on a Hilbert space H ⊂ H1(Ω). The eigenvalue problems are posed in the variational framework, with energy inner- product B(·, ·) and corresponding energy norm ||| · |||. Given an m-dimensional (m “small”) subspace Sˆ of approximate eigenfunctions in the (p or hp) finite element space V , our error estimates are based on the sines of the m principle angles, as measured in the energy inner-product, between Sˆ and an associated space S˜. These quantities, which we call approximation defects, provide an ideal measure of how “far” Sˆ is from being invariant in H, and are used for assessing both eigenvalue and invariant subspace approximation errors. Computable estimates of the approximation defects are obtained from an associated family of boundary value problems by means of auxiliary subspace error estimation: given u ∈ H and its B-projectionu ˆ ∈ V , an approximate error function ε ≈ u − uˆ is computed as the B-projection of u−uˆ onto an auxiliary subspace W —in the manner of traditional hierarchical basis error estimation, but with potentially more exotic choices of W . We discuss appropriate choices of W based on V , and provide corresponding reliability analysis; efficiency is trivial in this context. Experiments employing high-order polynomial spaces on meshes with both trian- gular and quadrilateral elements illustrate the practical performance, in terms of con- vergence and effectivities, of the proposed method.

9-7 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

KATO’S SQUARE ROOT THEOREM AS A BASIS FOR RELATIVE ESTIMATION THEORY OF EIGENVALUE APPROXIMATIONS Stefano Giani1, Luka Grubiˇsi´c2, Agnieszka Miedlar3 and Jeffrey S Ovall4

1School of Engineering and Computing Sciences, University of Durham, UK [email protected] 2Department of Mathematics, University of Zagreb, Croatia [email protected] 3Technische Universit¨atBerlin, Institut f¨urMathematik, Germany [email protected] 4Department of Mathematics, University of Kentucky, USA [email protected]

We present new residual estimates based on Kato’s square root theorem for spec- tral approximations of diagonalizable non-self-adjoint differential operators of diffusion- convection-reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations. We present a posteriori error estimates both for eigenvalues as well as eigenfunctions and prove that they are reliable. We demonstrate the efficiency of the proposed approach on a collection of benchmark examples.

9-8 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

HIGH PRECISION VERIFIED EIGENVALUE ESTIMATION FOR ELLIPTIC DIFFERENTIAL OPERATOR OVER POLYGONAL DOMAIN OF ARBITRARY SHAPE Xuefeng Liu

Research Institute for Science and Engineering, Waseda University, Japan [email protected]

This talk aims to propose a framework to provide high precision verified bounds for the leading eigenvalues of the self-adjoint elliptic differential operator over polygonal domain. This framework is based on several fundamental preceding research results [1-4]. To deal with the singularity of eigen-function in the case that the domain has an re-entrant corner, the method given in [1] is adopted. The high precision bounds are ob- tained by applying the Lehmann-Goerisch theorem [3,4] and the homotopy method[2] along with HP-FEM. Such kind of high precision eigenvalue estimation can be used to give sharp bounds for the interpolation error constants. It can also help to investigate the solution existence for boundary value problems of semi-linear elliptic differential equations.

References

[1] X. Liu, S. Oishi, Verified eigenvalue evaluation for Laplacian over polygonal do- main of arbitrary shape, to appear in SIAM Journal on Numerical Analysis, 2013.

[2] M. Plum, Bounds for eigenvalues of second-order elliptic differential operators, The Journal of Applied Mathematics and Physics(ZAMP), 42(6):848-863, 1991.

[3] N.J. Lehmann, Optimale eigenwerteinschließungen, Numerische Mathematik , 5(1):246-272, 1963.

[4] H. Behnke, F. Goerisch, Inclusions for eigenvalues of selfadjoint problems, Topics in Validated Computations (ed.J. Herzberger), North-Holland, Amsterdam, pp.277-322, 1994.

9-9 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

SPECTRAL ANALYSIS FOR A MIXED FINITE ELEMENT FORMULATION OF THE ELASTICITY EQUATIONS Salim Meddahi1, David Mora2 and Rodolfo Rodr´ıguez3

1 Departamento de Matem´aticas, Facultad de Ciencias, Universidad de Oviedo, Oviedo, Spain. [email protected] 2Departamento de Matem´atica, Universidad del B´ıo-B´ıo,Concepci´on,Chile. [email protected] 3CI2MA, Departamento de Ingenier´ıaMatem´atica, Universidad de Concepci´on,Concepci´on,Chile. [email protected]

This work deals with the approximation of the linear elasticity eigenvalue problem formulated in terms of the stress tensor and the rotation. This is achieved by con- sidering a mixed variational formulation in which the symmetry of the stress tensor is imposed weakly. We show that a discretization of the mixed eigenvalue elasticity problem with reduced symmetry based on the lowest order Arnold-Falk-Winther ele- ment, provides a correct approximation of the spectrum and prove quasi-optimal error estimates. Finally, we report some numerical experiments.

FINITE ELEMENTS FOR ELLIPTIC EIGENVALUE PROBLEMS IN THE PREASYMPTOTIC REGIME Stefan A. Sauter

Institut f¨urMathematik, Universit¨atZ¨urich, Winterthurerstr 190, CH-8057 Z¨urich, Switzerland [email protected]

Convergence rates for finite element discretisations of elliptic eigenvalue problems in the literature usually are of the form: If the mesh width h is fine enough then the eigenvalues resp. eigenfunctions converge at some well-defined rate. In our talk, we will analyse the maximal mesh width h0 - more precisely the minimal dimension of a finite element space - so that the asymptotic convergence estimates hold for h < h0. This mesh width will depend on the size and spacing of the exact eigenvalues, the spatial dimension and the local polynomial degree of the finite element space. We will show the results of some numerical experiments concerning a) the convergence of the eigenfunctions and - values, b) the convergence of the eigenvalue multigrid method to investigate the sharpness of the theoretical results. This work is in collaboration with L. Banjai and S. B¨orm.

9-10 9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

ACCURATE COMPUTATIONS OF MATRIX EIGENVALUES WITH APPLICATIONS TO DIFFERENTIAL OPERATORS Qiang Ye

Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, USA [email protected]

For matrix eigenvalue problems arising in discretizations of differential operators, it is usually smaller eigenvalues that well approximate the eigenvalues of the differential operators and are of interest. However, for ill-conditioned matrices, smaller eigenval- ues computed are expected to have low relative accuracy. In this talk, we present our recent works on high relative accuracy algorithms for computing eigenvalues of diago- nally dominant matrices. We present an algorithm that computes all eigenvalues of a symmetric diagonally dominant matrix to high relative accuracy. We further consider using the algorithm in an iterative method for a large scale eigenvalue problem and we show how smaller eigenvalues of finite difference discretizations of differential operators can be computed accurately.

9-11 10: Mini-Symposium: Error Estimation and adaptive modelling

10 Mini-Symposium: Error Estimation and adap- tive modelling

Organisers: Paul T Bauman and Kris van der Zee

10-1 10: Mini-Symposium: Error Estimation and adaptive modelling

REDUCED BASIS FINITE ELEMENT HETEROGENEOUS MULTISCALE METHOD FOR QUASILINEAR PROBLEMS Yun Bai

ANMC-MATHICSE-SB, Ecole´ Polytechnique F´ed´eralede Lausanne, Switzerland [email protected]

In this talk, we introduce a new multiscale method for quasilinear homogenization prob- lems, that combines the finite element heterogeneous multiscale method (FE-HMM) with reduced basis (RB) techniques based on an offline-online strategy. The FE-HMM for quasilinear multiscale problems [1], relies on a large number of micro problems that need to be computed in each iteration of the Newton method. As in addition macro and micro meshes need to be refined simultaneously, the FE-HMM for quasilinear problem can be costly. In contrast in the RB-FE-HMM, only a small number of micro problems selected by a rigorous a posteriori error estimator need to be computed [2]. We show that thanks to the new a posteriori error estimator, the result of [3] can be extended to quasilinear problem. A priori error estimates and convergence of the Newton method can be established. Joint work with A. Abdulle and G. Vilmart.

References

[1] A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems. To appear in Math. Comp., 2013.

[2] A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous mul- tiscale method for quasilinear elliptic homogenization problems. Preprint, submitted for publication, 2013.

[3] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems. J. Comput. Phys., 231(21) (2012) 7014-7036.

10-2 10: Mini-Symposium: Error Estimation and adaptive modelling

ERROR ESTIMATION AND ADAPTIVE MODELING FOR VISCOUS INCOMPRESSIBLE FLOWS Paul T. Bauman

Institute for Computational Engineering and Sciences The University of Texas at Austin 201 E. 24th St., Stop C0200, Austin, TX 78712, USA [email protected]

In this work, we consider a surrogate model to approximate solutions to the steady, incompressible Navier-Stokes equations. The surrogate model is constructed by replac- ing the Navier-Stokes model by the Stokes model in some region of the computational domain and by coupling the two models along an interface. Modeling error is incurred due to the approximation. We construct an estimate of the error, based on quantities of interest, and an adaptive modeling strategy to reduce the error by adjusting the position of the interface between the two models. We present two-dimensional numer- ical experiments to demonstrate the effectiveness of both the error estimator and the adaptive strategy. This is joint work with Timo van Opstal, Serge Prudhomme, and Harald van Brummelen.

GOAL-ORIENTED ERROR ESTIMATION AND ADAPTIVITY FOR THE TIME-DEPENDENT LOW-MACH NAVIER-STOKES EQUATIONS Varis Careya and Paul T. Bauman

Institute for Computational Engineering and Sciences, The University of Texas at Austin 201 E. 24th St., Stop C0200, Austin, TX 78712, USA [email protected]

We present a goal-oriented algorithm for error control as well as spatial, temporal and model adaptivity, targeting the low-mach compressible Navier-Stokes equations. The algorithm, using the GRINS computational framework, is illustrated for both stationary and non-stationary problems. Issues related to stabilization, linearization, and model choice are highlighted in the former case, while the additional interplay between storage, efficiency, and numerical accuracy of the forward and adjoint solutions is examined in non-stationary case.

10-3 10: Mini-Symposium: Error Estimation and adaptive modelling

ADAPTIVE INEXACT NEWTON METHODS WITH A POSTERIORI STOPPING CRITERIA FOR NONLINEAR DIFFUSION PDES Alexandre Ern1 and Martin Vohral´ık2

1Universit´eParis-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne la Vall´eecedex 2, France [email protected] 2INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France [email protected]

We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. To solve these systems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates also yield a guaranteed upper bound on the overall error at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework. We show how to apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the p-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach. More details on the overall approach, analysis, and results can be found in [1, 2].

References

[1] A. Ern and M. Vohral´ık, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. HAL Preprint 00681422 v2, 2012.

[2] A. Ern and M. Vohral´ık, Adaptive inexact Newton methods: a posteriori error control and speed-up of calculations, SIAM News, 46(1), 1, 2013.

10-4 10: Mini-Symposium: Error Estimation and adaptive modelling

TREE APPROXIMATION VERSUS AFEM Francesca Fierro1, Alfred Schmidt2 and Andreas Veeser3

1Dipartimento di Matematica, Universit`adegli Studi di Milano, Italy, [email protected], 2Zentrum f¨urTechnomathematik, Universit¨atBremen, Germany, [email protected], 3Dipartimento di Matematica, Universit`adegli Studi di Milano, Italy, [email protected]

Adaptive finite elements methods (AFEM) are an efficient tool for the solution of partial differential equations. We consider the so-called h-variant and assume that bisection is used for the mesh refinement. In this case, if the exact solution is known, the tree approximation algorithm of P. Binev and R. DeVore offers the opportunity to compute near best meshes at a cost that is linear in the number of bisections. After deriving a simpler local error indicator for this algorithm, it is easy to implement whenever bisection is available. In this talk we will report on numerical studies of AFEM that use the aforemen- tioned near best meshes as benchmarks.

10-5 10: Mini-Symposium: Error Estimation and adaptive modelling

CONTRACTION AND OPTIMAL CONVERGENCE OF A GOAL-ORIENTED ADAPTIVE FINITE ELEMENT METHOD Ricardo H. Nochetto1, A.J. Salgado1 and K.G. van der Zee2

1Dept. Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA 2Multiscale Engineering Fluid Dynamics, Dept. Mechanical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands [email protected]

We focus on a goal-oriented adaptive finite element method for an elliptic boundary- value problem, following the estimation and marking strategy proposed by Mommer and Stevenson [1]. In their pioneering work, they proved that the adaptive algorithm converges with an optimal rate, that is, the error in the output quantity of interest converges at a rate which is twice that of the usual energy-norm rate. Several extensions of [1] have recently appeared, for example, a different marking strategy is considered in [2], while nonsymmetric elliptic problems are considered in [3]. In the current contribution, we reconsider the original goal-oriented adaptive algo- rithm, and we provide a novel contraction result. Based on this result we re-establish the proof of optimal convergence. Several numerical experiments are presented that support our findings.

References

[1] Mommer, Stevenson, A goal-oriented adaptive finite element method with conver- gence rates, SIAM J Numer Anal 47-2, (2009), pp 861-886

[2] Becker, Estecahandy, Trujillo, Weighted marking for goal-oriented adaptive finite element methods, SIAM J Numer Anal 49-6, (2011), pp 2451-2469

[3] Holst, Pollock, Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems, (2011), arXiv:1108.3660v3

10-6 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

11 Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

Organisers: Volker John, Petr Knobloch and Julia Novo

11-1 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

A COMPUTABLE ERROR BOUND FOR A 3-DIMENSIONAL CONVECTION-DIFFUSION-REACTION EQUATION Mark Ainsworth1, Alejandro Allendes2, Gabriel R. Barrenechea3 and Richard Rankin4

1 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA. Mark [email protected] 2 Departamento de Matem´atica, Universidad T´ecnicaFederico Santa Mar´ıa, Av. Espa˜na1680, Casilla 110-V Valpara´ıso,Chile. [email protected] 3 Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland. [email protected] 4 Computational and Applied Mathematics Department, Rice University, 6100 Main Street, MS-134 Houston, TX 77005-1892, USA. [email protected]

Fully computable upper bounds are developed for the discretisation error measured in the natural (energy) norm for convection-reaction-diffusion problems in three dimen- sions. The upper bounds are genuine upper bounds in the sense that the numerical value of the estimated error exceeds the actual numerical value of the true error re- gardless of the coarseness of the mesh or the nature of the data for the problem. All constants appearing in the bounds are fully specified. Examples show the estimator to be reliable and accurate even in the case of complicated three dimensional problems.

11-2 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

AUGMENTED TAYLOR-HOOD ELEMENTS FOR INCOMPRESSIBLE FLOW Daniel Arndt

Institute for Numerical and Applied Mathematics, University of Goettingen, Germany. [email protected]

It is well-known that in finite element discretizations of incompressible flow problems the numerical solution is in general not pointwise solenoidal, unless the divergence of the velocity ansatz space is contained in the pressure ansatz space. Although many turbulence models are based on the conservation of mass, in discretizations often finite elements are used that are not divergence free. In this talk, we consider an augmented Taylor-Hood pair that improves conservation of mass. The modification is given by adding elementwise constant functions to the pressure space. We examine the inf-sup stability of this discretization for quadrilateral and hexahedral meshes. Furthermore, convergence results for a Stokes problem with discontinuous pressure are presented. As a test case for the Navier-Stokes equations we consider a turbulent channel flow and examine how the improved mass conservation influences the quality of the numerical solution using a turbulence model by Verstappen.

11-3 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

A NONLINEAR DISSIPATION TO AVOID LOCAL OSCILLATIONS FOR THE FINITE ELEMENT APPROXIMATION OF THE CONVECTION-DIFFUSION EQUATION Joan Baigesa and Ramon Codinab

Universitat Polit`ecnicade Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain [email protected], [email protected]

When diffusion is small compared to convection, the Galerkin finite element approxi- mation of the convection-diffusion equation suffers from numerical instabilities. This is a well known and, in fact, possibly the first concept one learns when approximating flow problems using finite elements. Several remedies have been devised along the years to overcome this problem, starting with the introduction of purely artificial dissipation in the von Neumann line and leading to several stabilization methods used nowadays. We also employ one of such methods, which consists in adding a stabilizing term to the Galerkin ones depending on the residual of the equation to be solved. Our formulation can be framed within the variational multi scale concept introduced by T.J.R. Hughes in 1995. Stabilization methods, and in particular the one we favor, are intended to provide global stability to the discrete finite element problem. This in particular means that one can show stability and convergence of the discrete solution in global norms, i.e., norms that involve the L2 norm of some terms over all the computational domain. The global instabilities displayed by the Galerkin approximation are thus avoided but, nevertheless, local oscillations may still remain and, in fact, do appear in the neigh- borhood of sharp gradients of the discrete solution. This is due to the fact that local control and particularly control in the L∞ norm cannot be achieved. In order to avoid the appearance of local instabilities several methods have also been proposed over the years. In essence, they all rely on the introduction of a nonlinear dissipation close to where there are sharp gradients of the approximate solution. These dissipations can be motivated in different ways, physical or analytical, in the form of explicit dissipation or by introducing limiters to the discrete solution. The aim of this work is to present a nonlinear dissipation to avoid local oscillations in the sense described. Succinctly, it has the form of the artificial dissipation that guarantees that no oscillations may appear but multiplied by a factor that depends on the discrete solution, thus making the formulation nonlinear. This factor is taken as the projection orthogonal to the finite element space of the gradient of the discrete unknown, normalized by the gradient itself. After describing the formulation, we show that the resulting nonlinear problem is well posed and present several numerical ex- amples which show that the method succeeds in removing local oscillations, nonlinear convergence is satisfactory and is less over diffusive far from sharp layers than other methods designed with the same objective.

11-4 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

INVESTIGATIONS OF A FEM-FCT SCHEME APPLIED TO A 1D MODEL PROBLEM Gabriel R. Barrenechea1, Volker John2 and Petr Knobloch3

1Department of Mathematics and Statistics, University of Strathclyde, Glasgow, Scotland, [email protected], 2WIAS Berlin and Free University of Berlin, Germany, [email protected], 3Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic, [email protected]

It is well known that Galerkin finite element discretizations are not appropriate for the numerical solution of convection dominated problems since the approximate solutions are typically globally polluted by spurious oscillations of unacceptable magnitude, see, e.g., [3]. Among various remedies, the FEM-FCT schemes proved to be rather efficient [1]. However, for this type of methods, there are no theoretical investigations concern- ing existence, uniqueness and convergence of approximate solutions available in the literature. We consider the algebraic flux correction scheme described in [2] and apply it to a steady one-dimensional convection–diffusion equation. This leads to a nonlinear differ- ence scheme whose properties are investigated in detail. In particular, we demonstrate that this problem is generally not solvable and we discuss various improvements.

References

[1] V. John and E. Schmeyer. Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion. Comput. Methods Appl. Mech. Engrg., 198:475–494, 2008.

[2] Dmitri Kuzmin. Algebraic flux correction for finite element discretizations of cou- pled systems. In M. Papadrakakis, E. O˜nate,and B. Schrefler, editors, Proceedings of the Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering, pages 1–5. CIMNE, Barcelona, 2007.

[3] H.-G. Roos, M. Stynes, and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion–Reaction and Flow Prob- lems. 2nd ed. Springer-Verlag, Berlin, 2008.

11-5 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

A POSTERIORI ERROR ESTIMATION IN STABILIZED DISCRETIZATIONS OF STATIONARY CONVECTION-DIFFUSION-REACTION PROBLEMS Markus Bausea and Kristina Schweglerb

Faculty of Mechanical Engineering, Helmut Schmidt University, University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany [email protected], [email protected]

The reliable numerical approximation of convection-diffusion-reaction problems

b · ∇u − ∇ · (A∇u) + r(u) = f (1) with small diffusion A is still a challenging task. Eq. (1) is considered as a prototype model for more sophicated equations of practical interest. For its numerical solution stabilized methods are used that aim to introduce a correct amount of artificial diffusion in regions with sharp inner or boundary layers or complicated structures where impor- tant phenomena take place; cf., e.g., [M. Bause, K. Schwegler, Analysis of stabilized higher-order finite element approximation of nonstationary and nonlinear convection- diffusion-reaction equations, Comput. Methods Appl. Mech. Engrg., 209–212 (2012), 184–196]. In this work we consider using lower and higher order conforming finite element methods with streamline upwind Petrov-Galerkin (SUPG) stabilization. In addition, discontinuity- or shock capturing stabilization as an additional consistent modifica- tion of the numerical scheme and stabilization in crosswind direction is applied. To further improve the approximation quality and the efficiency of the calculations, we combine the stabilized discretization with an a posteriori error control mechanism and an adaptive mesh generation algorithm based on a dual-weighted-residual approach. The dual weighted error estimator assesses the discretization error with respect to a given quantity of physical interest. We study different approaches for combining SUPG and shock-capturing stabiliza- tion with dual-weighted-residual error estimation. Various output functionals of the solution are proposed. By numerical experiments we analyze and illustrate the effi- ciency and performance properties of the algorithms.

11-6 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

ROBUST ERROR ESTIMATES IN WEAK NORMS WITH APPLICATION TO IMPLICIT LARGE EDDY SIMULATION Erik Burman

Department of Mathematics, University College London, London, U.K. [email protected]

In this talk we will discuss a posteriori and a priori error estimates of filtered quantities for solutions to some equations of fluid mechanics. For the computation of the solution we use low order finite element methods using either linear or nonlinear stabilization. To obtain estimates that are robust with respect to the diffusion/viscosity coefficient we introduced a class of weak norms corresponding to taking a weighted H1-norm of a filtered solution. For these weak norms we propose error estimates whose error constants depend only on the regularity of the initial data. In particular the estimates are independent of the Reynolds number, the Sobolev norm of the exact solution at time t > 0, or nonlinear effects such as shock formation. It follows that we obtain a complete assessment of the computability of the solution given the initial data. After a detailed description of the analysis in the case of the Burgers’ equation we widen the scope and discuss two dimensional incompressible turbulence and passive transport with rough data within the same paradigm.

11-7 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

ANISOTROPIC LOCAL PROJECTION STABILIZATION IN STREAMLINE AND CROSSWIND DIRECTIONS Helene Dallmanna and Gert Lubeb

Institute for Numerical and Applied Mathematics, Georg-August University of G¨ottingen,Germany [email protected], [email protected]

The local projection stabilization (LPS) method splits the discrete ansatz spaces into small and large scales and adds stabilization terms only on the small ones. In [2] an analysis for finite element discretizations of linearized incompressible flows using the local projection method is given. Here stabilizing terms for the streamline direc- tion of the velocity gradient, for the incompressibility constraint and the pressure are used. Furthermore, [1] also studies stabilizations in the crosswind direction for scalar equations. We consider an anisotropic stabilization technique for the vector valued Oseen case and present an analysis for the resulting problem; stabilizing terms in streamline and crosswind direction are introduced into the equations. In addition, we look into the generalization to the nonlinear incompressible Navier Stokes model.

References

[1] Gabriel R Barrenechea, Volker John, and Petr Knobloch. A Local Projection Stabi- lization Finite Element Method with Nonlinear Crosswind Diffusion for Convection- diffusion-reaction Equations. WIAS, 2012.

[2] G. Lube, G. Rapin, and J. L¨owe. Local projection stabilization for incompress- ible flows: Equal-order vs. inf-sup stable interpolation. Electronic Transactions on Numerical Analysis, 32:106–122, 2008.

11-8 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

ON SUPERCONVERGENCE FOR HIGHER-ORDER FEM IN CONVECTION-DIFFUSION PROBLEMS Sebastian Franz

Institute for Numerical Mathematics, TU Dresden, Germany [email protected]

For singularly perturbed convection-diffusion problems and many numerical methods a supercloseness property is known for bilinear elements. This means that the difference between the numerical solution uN , obtain by a Galerkin FEM or a stabilised FEM, and the bilinear interpolant of the exact solution u is convergent of order two in the energy norm, although uN − u is only convergent of order one. We will investigate similar properties for higher-order FEM and look especially at the choice of suitable interpolation operators and results for stabilised methods. Having a supercloseness property, it is cheap to obtain a better numerical solution by simple postprocessing — a superconvergent solution. We will also address different possibilities for postprocessing.

AN ADAPTIVE SUPG METHOD FOR EVOLUTIONARY CONVECTION-DIFFUSION EQUATIONS Javier de Frutos1, Bosco Garc´ıa-Archilla2 and Julia Novo3

1IMUVA, Unversidad de Valladolid, Spain. [email protected] Departamento de Matem´aticaAplicada II, Universidad de Sevilla, Spain. [email protected] Departamento de Matem´aticas, Universidad Aut´onomade Madrid, Spain. [email protected]

In [V. John & J. Novo, A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations, CMAME, 2013, 289-305] a robust residual-based a pos- teriori estimator for the SUPG finite element method applied to stationary convection- diffusion problems is proposed. In this work we extend this residual estimator to evolutionary convection-diffusion equations. The main idea is that the SUPG approx- imation to the evolutionary problem is also the SUPG approximation to a particular steady convection-diffusion problem with right-hand side depending on the computed approximation. Based on the a posteriori error estimator an adaptive algorithm is de- veloped. Some numerical experiments are shown in which the new adaptive procedure compares favourably with the adaptive method based on the standard Galerkin finite element method proposed in [J. de Frutos, B. Garc´ıa-Archilla & J. Novo, CMAME, (2011), 3601-3612.]

11-9 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

SUPG FINITE ELEMENT METHOD FOR PDES IN TIME-DEPENDENT DOMAINS Sashikumaar Ganesana and Shweta Srivastavab

Numerical Mathematics and Scientific Computing, SERC, Indian Institute of Science, Bangalore 560012, India. [email protected], [email protected]

The numerical solution of convection dominated convection–diffusion problems is one of the challenging and active research fields. It is well known that the standard Galerkin finite element methods induce spurious oscillations in the numerical solution of these problems. Streamline–Upwind–Petrov–Galerkin (SUPG) is one of the popular sta- bilization methods proposed for steady-state convection dominated problems in [3]. Recently, it has been analyzed for time-dependent scalar equations in [1,2,4]. In this talk, the SUPG finite element method for a time-dependent scalar equa- tion in a time-dependent domain will be presented. Apart form the other challenges associates with the solution of convection dominated problems, the time-dependent domain makes the problem more challenging. We handle the deformation of the do- main with the arbitrary Lagrangian–Eulerian (ALE) approach. The conservative and non-conservative ALE form of the scalar problem will be discussed. Further, the anal- ysis of the SUPG applied to the conservative ALE form of the scalar problem will be presented. Finally, the numerical results for an array of problems will be presented.

References

[1] E. Burman: Consistent SUPG-method for transient transport problems: Stability and convergence, Comput. Methods in Appl. Mech. and Engrg., 199, (2010) 1114– 1123.

[2] S. Ganesan: An operator-splitting Galerkin/SUPG finite element method for pop- ulation balance equations: Stability and convergence, ESAIM: Mathematical Mod- elling and Numerical Analysis (M2AN), 46, (2012) 1447–1465.

[3] T.J.R. Hughes and A.N. Brooks: A multi-dimensional upwind scheme with no cross- wind diffusion, In Finite element methods for convection dominated flows, T.J.R. Hughes, ed., Vol AMD 34, ASME, New York (1979).

[4] V. John and J. Novo: Error Analysis of the SUPG Finite Element Discretization of Evolutionary Convection-Diffusion-Reaction Equations, SIAM J. Numer. Anal., 49, (2011) 1149–1176.

11-10 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

STABILIZATION OF CONVECTION-DIFFUSION PROBLEMS BY SHISHKIN MESH SIMULATION Bosco Garc´ıa-Archilla

Depto. de Matem´aticaAplicada II, Universidad de Sevilla, Spain. [email protected]

We present a new stabilization procedure for numerical methods for convection-diffusion problems. It is based on a simulation of the interaction between the coarse and fine parts of a Shishkin grid, but it can be easily applied on coarse and irregular meshes and on domains with nontrivial geometries. The technique, which does not require adjust- ing any parameter, can be applied to different stabilized and non stabilized methods. Numerical experiments show it to obtain oscillation-free approximations on problems with boundary and internal layers, on uniform and nonuniform meshes and on domains with curved boundaries.

A ROBUST SUPG NORM A POSTERIORI ERROR ESTIMATOR FOR STATIONARY CONVECTION-DIFFUSION EQUATIONS Volker John1 and Julia Novo2

1Weierstrass Institute for Applied Analysis and Stochastics, WIAS, Berlin, Germany. [email protected] 2Departamento de Matem´aticas,Universidad Aut´onomade Madrid, Spain. [email protected]

A robust residual-based a posteriori estimator is proposed for the SUPG finite ele- ment method applied to the stationary convection-diffusion-reaction equations. The error in the natural SUPG norm is estimated. The main concern of the paper is the consideration of the convection-dominated regime. A global upper bound and a local lower bound for the error are derived, where the global upper estimate relies on some hypothesis. Numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypothesis. A comparison to other residual-based estimators with respect to the adaptive grid refinement is also provided.

11-11 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

VELOCITY-PRESSURE REDUCED ORDER MODELS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS Volker John

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany, [email protected] and Free University of Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany

Reduced order modeling (ROM) uses global basis functions that were derived from simulations with standard discretizations or even from experimental data. In this way, only a small number of basis functions is necessary for performing simulations that provide the most important features of the solution. ROM is already often used in op- timization, where repeated simulations of a problem with (slightly) varying parameters are necessary. In the case of the incompressible Navier–Stokes equations, mostly only a veloc- ity ROM is performed. However, the pressure is often of importance, e.g., for the computation of functionals of interest. In this talk, three velocity-pressure ROMs from two classes will be introduced. One of the methods seems to be new. The methods will be compared in numerical studies of a laminar flow around a cylinder, which consider, in particular, the drag and lift coefficient at the cylinder.

A FINITE ELEMENT METHOD FOR A NONCOERCIVE ELLIPTIC CONVECTION DIFFUSION PROBLEM Klim Kavalioua and Lutz Tobiskab

Faculty of Mathematics, Otto von Guericke University, Magdeburg, Germany. [email protected], [email protected]

A combined finite-element finite-volume method is applied on a noncoercive elliptic boundary value problem. The method is based on triangulations of weakly acute type and a secondary circumcentric subdivision. The properties of the continuous problem, that the kernel is one-dimensional and spanned by a positive function, are preserved in the discrete case. A priori error estimates of first order in the H1-norm are shown for sufficiently small mesh sizes. Numerical test examples confirm the theoretical predictions.

11-12 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

ON THE ROLE OF THE HELMHOLTZ DECOMPOSITION IN MIXED METHODS FOR INCOMPRESSIBLE FLOWS AND A NEW VARIATIONAL CRIME Alexander Linke

Department of Mathematics and Computer Science, Free University Berlin, Arnimallee 3, 14195 Berlin, Germany, [email protected]

In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations should be balanced completely by the pressure gradient. Unfortunately, nearly all available discretizations for incompressible flows violate this property. The origin of the problem is that discrete velocities are usually not divergence-free. Hence, the use of divergence-free velocity reconstructions is proposed wherever an L2 scalar product appears in the discrete variational formulation, which actually means committing a variational crime. The approach is illustrated and applied to a nonconforming Crouzeix-Raviart finite element discretization. It will be proved and numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart-Thomas element increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.

11-13 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

A TWO-LEVEL LOCAL PROJECTION STABILISATION ON UNIFORMLY REFINED TRIANGULAR MESHES Gunar Matthies1 and Lutz Tobiska2

1Fachbereich Mathematik und Naturwissenschaften, Institut f¨urMathematik, Universit¨atKassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany [email protected] 2Institut f¨urAnalysis und Numerik, Otto-von-Guericke-Universit¨atMagdeburg, PSF 4120, 39016 Magdeburg, Germany [email protected]

The local projection stabilisation (LPS) has been successfully applied to scalar conv- ection-diffusion-reaction equations, the Stokes problem, and the Oseen problem. A fundamental tool in its analysis is that the interpolation error of the approxima- tion space is orthogonal to the discontinuous projection space. It has been shown that a local inf-sup condition between approximation space and projection space is sufficient to construct modifications of standard interpolations which satisfy this additional or- thogonality. There are different versions of the local projection stabilisation on the market; we will consider the two-level approach based on standard finite element spaces Yh on a mesh Th and on projection spaces Dh living on a macro mesh Mh. Hereby, the finer mesh is generated from the macro mesh by a certain refinement rules. In the usual two- level local projection stabilisation on triangular meshes, each macro triangle M ∈ Mh is divided by connecting its barycentre with its vertices. Three triangles T ∈ Th are disc obtained. Then, the pairs (Pr,h,Pr−1,2h), r ≥ 1, of spaces of continuous, piecewise polynomials of degree r on Th and discontinuous, piecewise polynomials of degree r − 1 on Mh satisfy the local inf-sup condition and can be used within the LPS framework.

One disadvantage of this refinement technique is however that Th contains simplices with large inner angles even in the case of a uniform decomposition Mh into isosceles triangles. Another drawback is that this refinement rule leads to non-nested meshes and spaces whereas the common refinement technique of one triangle into 4 similar triangles (called red refinement in adaptive finite elements) results into nested meshes and spaces. disc We will show that in the two-dimensional case the pairs (Pr,h,Pr−1,2h), r ≥ 2, satisfy the local inf-sup condition with the refinement of one triangle into 4 triangles. Con- sequently, the LPS can be also applied on sequences of nested meshes and spaces and keeping the same error estimates. Finally, we compare the properties of the two result- ing LPS methods based on the different refinement strategies by means of numerical test examples for convection-diffusion problems with dominating convection.

11-14 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

A FLUX-CORRECTED TRANSPORT METHOD BASED ON LOCAL PROJECTION STABILIZATION FOR NON-STATIONARY TRANSPORT PROBLEMS Friedhelm Schieweck1 and Dmitri Kuzmin2

1Institut f¨urAnalysis und Numerik, Otto-von-Guericke Universit¨atMagdeburg, Postfach 4120, D-39016 Magdeburg, Germany [email protected] 2Applied Mathematics III, University Erlangen-Nuremberg, Cauerstr. 11, D-91058, Erlangen, Germana [email protected]

The local projection stabilization (LPS) method is an attractive space discretization technique for non-stationary convection-dominated transport problems. Combined with an implicit time discretization, it yields a stable and non-oscillatory high-order finite element approximation in regions where the exact solution is sufficiently smooth. However, the accuracy of the LPS solution deteriorates in the neighborhood of dis- continuities or steep gradients. Like any other linear high-order scheme, LPS tends to produce spurious oscillations in these regions. The usual way to avoid such local oscillations is to add some shock-capturing terms acting as nonlinear artificial viscosity. In many cases, this fix leads to marked improvements but the results depend on the choice of a free parameter, which undermines the practical utility of such schemes. In contrast to traditional shock capturing, the flux-corrected transport (FCT) methodology makes it possible to prevent numerical oscillations and to enforce the discrete maximum principle in a fail-safe manner. FCT can be regarded as a technique for blending a stable high-order discretization with a monotone low-order discretiza- tion which contains enough artificial diffusion to suppress undershoots and overshoots. To this end, the difference between the two approximations is decomposed into anti- diffusive fluxes or element contributions which are multiplied by solution-dependent correction factors. The FCT solution is guaranteed to be non-oscillatory everywhere and to revert to the underlying high-order approximation in smooth regions. In the finite element literature, the FCT method has successfully been used to combine an unstable high-order Galerkin scheme with a low-order upwind-biased dis- cretization. Since the unconstrained Galerkin approximation may exhibit global oscil- lations, the FCT limiter may need to be activated everywhere, which may destroy the high accuracy in regions of smoothness. This concern has led us to combine the stable high-order LPS discretization with a low-order artificial viscosity method. Since the LPS method is linearly stable, the FCT correction of antidiffusive fluxes is restricted to small subdomains, whereas optimal accuracy is maintained elsewhere. In this talk, we discuss the practical implementation of the proposed FCT method in the case of one space dimension and present some numerical results.

11-15 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

TOWARDS ANISOTROPIC QUALITY TETRAHEDRAL MESH GENERATION Hang Si

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany [email protected]

Many physical problems exhibit anisotropic features, i.e., their solutions change more significantly in one direction than others. Examples include in particular convection- dominated problems whose solutions have, e.g., layers, shocks, or corner singularities. When numerical methods are used to approximate these problems, it is of great im- portance that the used meshes represent such features to achieve high accuracy at a low computational cost. Tetrahedral meshes are very popular for discretizing three-dimensional domains. They can be easily adapted to arbitrary geometries, they can be refined locally, and they can be generated automatically. However, the generation of anisotropic quality tetrahedral meshes is a complex problem and it is an active research topic. This talk addresses the state of the art in high quality tetrahedral mesh generation, presenting in particular a relative robust and efficient methodology implemented in the program TetGen for generating isotropic meshes, and it discusses important open questions which arise in generalizations to the anisotropic case.

11-16 11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

A LOCAL PROJECTION STABILIZATION METHOD FOR FINITE ELEMENT APPROXIMATION OF A MAGNETOHYDRODYNAMIC MODEL Benjamin Wackera and Gert Lubeb

Institute for Numerical and Applied Mathematics, University of G¨ottingen, G¨ottingen,Germany. [email protected], [email protected]

In this talk, we consider the equations of incompressible resistive magnetohydrody- namics. Based on a stabilized finite element formulation by S. Badia, R. Codina and R. Planas for the linearized equations [1], we propose a modification of this technique by a local projection stabilization finite element method for the approximation of this problem. The introduced stabilization technique is then discussed by investigating the sta- bility and convergence analysis for the problem’s formulation thoroughly. We finally compare our numerical analysis with other approximations presented in the literature. Finally, we give an outlook to the application of the approach to the fully nonlinear MHD model [2].

References

[1] S. Badia, R. Codina and R. Planas. On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics, Journal of Com- putational Physics, 234:399-416, 2013.

[2] D. Sondak, A. Oberai. Large eddy simulation models for incompressible magne- tohydrodynamics derived from the variational multiscale formulation. Physics of Plasmas, 19, 102308, 2012.

11-17 12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

12 Mini-Symposium: Finite Element Methods for Multiphysics Problems

Organisers: Norbert Heuer and Salim Med- dahi

12-1 12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

A STABILIZED FINITE VOLUME ELEMENT FORMULATION FOR SEDIMENTATION-CONSOLIDATION PROCESSES Raimund B¨urger1, Ricardo Ruiz-Baier2 and H´ectorTorres3

1 CI2MA and Departamento de Ingenier´ıaMatem´atica, Facultad de Ciencias F´ısicasy Matem´aticas, Universidad de Concepci´on,Concepci´on, Chile [email protected] 2 CMCS-MATHICSE-SB, Ecole Polytechnique F´ed´eralede Lausanne, Lausanne, Switzerland [email protected] 3 Departamento de Matem´aticas, Facultad de Ciencias, Universidad de La Serena, La Serena, Chile [email protected]

A model of sedimentation-consolidation processes in so-called clarifier-thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the Stokes system for an incompressible fluid describing the motion of the mixture. In cylindrical coordinates, and if an axially symmetric solution is assumed, the original problem reduces to two space dimensions. This poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted Sobolev spaces. A novel finite volume element method is introduced for the spatial discretization, where the velocity field and the solids concentration are discretized on two different dual meshes. The method is based on a stabilized discontinuous Galerkin formulation for the concentration field, and a multiscale stabilized pair of P1-P1 elements for velocity and pressure, respectively. In this presentation, numerical experiments illustrate properties of the model and the satisfactory performance of the proposed method.

12-2 12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

DOUBLE LAYER POTENTIAL BOUNDARY CONDITIONS FOR THE HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD Zhixing Fu1,a, Norbert Heuer2 and Francisco-Javier Sayas1,c

1Department of Mathematical Sciences, University of Delaware, Newark DE, USA [email protected], [email protected] 2Facultad de Matem´aticas,Pontificia Universidad Cat´olicade Chile, Santiago, Chile. [email protected]

In this talk we present a simple coupling strategy for the Discontinuous Galerkin Method with Boundary Element Methods. The coupling is based on a Galerkin dis- cretization of the second (or hypersingular) boundary integral equation sharing nu- merical flows with the HDG scheme. We first show how the system can be hybridized to be solved only on the skeleton of the triangulation and the coupling interface. We next give sufficient conditions on the diffusion parameter guaranteeing coercivity at the discrete level. Finally, we will show some numerical experiments confirming the theoretical finds and extending their applicability to situations where the coercivity threshold is crossed.

12-3 12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

A LINEAR FINITE ELEMENT SCHEME FOR THE STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION Beniamin Goldys1, Kim-Ngan Le2b and Thanh Tran2c

1School of Mathematics and Statistics, The University of Sydney, Sydney 2006, Australia [email protected] 2School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia [email protected], [email protected]

The study of the theory of ferromagnetism involves the Landau–Lifshitz–Gilbert equa- tion. Since stationary solutions are in general not unique, it is important to describe phase transitions between different equilibrium states induced by thermal fluctuations of the effective field. This is done by adding noise to the effective field. The equation then takes the form
dM = λ1M × ∆M − λ2M × (M × ∆M) dt + (M × g) ◦ dW (t),

3 where λ1 6= 0 and λ2 > 0 are constants, and g : D → R is a given bounded function, with D being a bounded domain in Rd, d = 2, 3, having smooth boundary ∂D. Here ◦ dW (t) stands for the Stratonovich differential of the Wiener process W (t). The unknown function M : [0,T ] × D → R3 satisfies the following conditions ∂M(t, x) = 0 ∀t ∈ (0,T ) and x ∈ ∂D, ∂n M(0, x) = M0(x) ∀x ∈ D, |M(t, x)| = 1 ∀t ∈ [0,T ] and x ∈ D.

In this talk we present a linear finite element scheme for the problem and show that a weak martingale solution exists. Numerical experiments confirm our theoretical results.

12-4 12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

A DECOUPLED PRECONDITIONING TECHNIQUE FOR A MIXED STOKES-DARCY MODEL Antonio M´arquez1, Salim Meddahi2 and Francisco-Javier Sayas3

1Departamento de Construcci´on,Universidad de Oviedo, Spain [email protected] 2Departamento de Matem´aticas, Universidad de Oviedo, Spain [email protected] 3Department of Mathematical Sciences, University of Delaware, USA, [email protected]

Our aim is to provide an efficient iterative method to solve the mixed Stokes-Dracy model for coupling fluid and porous media flow. We consider for this model a formu- lation relying on an H(div)-approach in the Darcy domain. The Stokes problem is expressed in the usual velocity-pressure form. The resulting weak formulation leads to a coupled, indefinite, ill-conditioned and symmetric linear system of equations. Opti- mal iterative methods are then important for solving efficiently these linear equations. Ideally, the algorithm should uncouple the global model in such a way that, only independent Stokes and Darcy subproblems are involved at each iteration step. We in- troduce a decoupled iterative process consisting in two nested MINRES methods whose preconditioners only require the solution of several second-order H1-elliptic problems in the Stokes and the Darcy domains. Theoretical analysis and numerical experiments show the optimality and efficiency of the proposed decoupled iterative solver.

CONFORMING AND DIVERGENCE-FREE STOKES ELEMENTS Michael Neilan

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA [email protected]

A family of finite elements methods for the velocity-pressure formulation of the Stokes equations are developed. In our approach we enrich H(div; Ω)-conforming finite ele- ments locally with divergence-free rational bubble functions to enforce strong tangental continuity while still preserving the inf-sup condition of the original H(div; Ω) finite element space. We show that the method converges optimally on general shape-regular triangulations, and that the velocity error is completely decoupled from the pressure error. Moreover, the pressure space is exactly the image of the divergence operator act- ing on the velocity space. Therefore, the discretely divergence-free velocity functions are divergence-free pointwise. We also show how the proposed elements are related to a class of C1 Zienkiewicz-type finite elements through the use of a smooth discrete de Rham (Stokes) complex.

12-5 12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

AN EXACTLY DIVERGENCE-FREE FINITE ELEMENT METHOD FOR A GENERALIZED BOUSSINESQ PROBLEM Ricardo Oyarz´ua1 and Dominik Sch¨otzau2

1Departmento de Matem´atica, Universidad del B´ıo-B´ıo,Concepci´on,Chile, [email protected], 2Mathematics Department, University of British Columbia, Canada [email protected]

In this talk we present a mixed finite element method with exactly divergence-free ve- locities for the numerical simulation of a generalized Boussinesq problem, describing the motion of a non-isothermal incompressible fluid subject to a heat source. The method is based on using divergence-conforming elements of order k for the velocities, discon- tinuous elements of order k − 1 for the pressure, and standard continuous elements of order k for the discretization of the temperature. The H1-conformity of the velocities is enforced by a discontinuous Galerkin approach. The resulting numerical scheme yields exactly divergence-free velocity approximations; thus, it is provably energy-stable with- out the need to modify the underlying differential equations. We prove the existence and stability of discrete solutions, and derive optimal error estimates in the mesh size for small and smooth solutions.

HP-TIME-DISCONTINUOUS GALERKIN FOR PRICING AMERICAN PUT OPTIONS Ernst P. Stephan

Institute for Applied Mathematics, Leibniz Universit¨atHannover, Hannover, Germany [email protected]

The time-discontinuous Galerkin hp-finite element method is applied to the Black- Scholes partial differential equation for American put options. The first approach exploits a weak non-penetration condition and a Lagrange multiplier space which is spanned by biorthogonal basis functions. The arising problem is solved by a globalized semi-smooth Newton (SSN) method with a penalized Fischer-Burmeister non-linear complementarity function. It is shown that the reduced SSN method converges locally Q-quadratic. The second, equivalent approach relaxes the non-penetration condition to a discrete set of Gauss-Lobatto points in space and time which is incorporated in the ansatz and test space yielding a non-symmetric linear variational inequality. Numerical examples confirm the superiority of the two hp-approaches in terms of error reduction and computational time and of the mixed hp-method, i.e. the first approach, in particular.

12-6 13: Mini-Symposium: Finite Elements in Nonlinear Spaces

13 Mini-Symposium: Finite Elements in Nonlinear Spaces

Organisers: Philipp Grohs and Oliver Sander

13-1 13: Mini-Symposium: Finite Elements in Nonlinear Spaces

SUBDIVISION METHOD FOR THE CANHAN-HELFRICH MODEL Jingmin Chen1, Sara Grundel2, Robert Kusner3, Thomas Yu1a and Andrew Zigerelli1

1Department of Mathematics, Drexel University, Phiadelphia, U.S.A.. [email protected] 2Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany 3Department of Mathematics, University of Massachusetts at Amherst, Amherst MA 01003, U.S.A.

Lipid bilayers are ubiquitious in biological systems, and their equilibrium shapes are widely believed to be governed by the Canhan-Helfrich model. In the parametric FEM methods by Bonito/Nochetto/Pauletti, Deckelnick/Dziuk/Elliott, etc. for numerically solving this model, piecewise linear or piecewise quadratic elements are used for ap- proximating the membrane surface, and a technical weak formulation is derived. In this talk, we develop a different numerical method based on a technique from computer- aided geometric design known as subdivision surface. Unlike piecewise polynomial surfaces, subdivision surfaces have just enough regularity for us to directly and ac- curately compute their Willmore energy – the key functional in the curvature-based Canham-Helfrich model; thus no weak formulation is needed and the equilibrium shape can be computed based on a nonlinear optimization solver. We also discuss, among other nontrivial mathematical properties, some curious phase transition phenomena in the Canhan-Helfrich model observed based on our nu- merical method. Some of these were never addressed in either the biophysics or the geometric analysis communities.

13-2 13: Mini-Symposium: Finite Elements in Nonlinear Spaces

ON POTTS AND BLAKE-ZISSERMAN FUNCTIONALS FOR MANIFOLD-VALUED DATA Laurent Demareta, Martin Storathb and Andreas Weinmannc

Institute of Biomathematics and Biometry, Helmholtz Zentrum Munchen and Department of Mathematics, TU Munchen, Germany. [email protected], [email protected], [email protected]

For real-valued data Potts functionals are energy functionals of the form

2 Pγ(u) = γ · J(u) + ku − fk2.

Here the regularizing term J(u) counts the number of jumps of u whereas the fidelity to univariate discrete data is measured in the `2 norm. The family of Blake-Zisserman functionals is obtained by replacing the jump penalty J by some cut-off quadratic variation Js, i.e., X 2 2 Js(u) = min(s , |ui − ui−1| ), s > 0. i Minimization of these functionals is usually used for denoising and for (multi-label) segmentation tasks. If the data are not real-valued but take their values in a (connected) Riemannian manifold (which includes the case of a vector space), we may still define (univariate) manifold-valued Potts functionals by

X 2 Pγ(u) = γ · J(u) + dist(ui, fi) . i Here dist(·, ·) is the distance in the Riemannian manifold. Manifold-valued Blake- Zisserman functionals are obtained by replacing the data term likewise and by replacing the absolute values |ui − ui−1| in Js(u) by the distances dist(ui, ui−1). We obtain the existence of minimizers and an algorithm to compute minimizers for the class of Cartan-Hadamard manifolds. Those are complete Riemannian manifolds of nonpositive sectional curvature. Examples are the manifolds of positive matrices Posn which are the data space in diffusion tensor imaging, the hyperbolic and the euclidean spaces. Even for real-valued data, the multivariate Potts and Blake-Zisserman problems are NP-hard which means that we cannot expect to find a computationally feasible algorithm always yielding a minimizer. Accepting this fact we develop heuristic mini- mization strategies for multivariate manifold-valued data. We apply our algorithms to diffusion tensor data.

13-3 13: Mini-Symposium: Finite Elements in Nonlinear Spaces

B-SPLINE QUASIINTERPOLATION OF MANIFOLD-VALUED DATA Philipp Grohs

Seminar for Applied Mathematics, ETH Zurich, Switzerland [email protected]

We consider the problem of approximating manifold-valued functions with approxima- tion spaces spanned by linear combinations of cardinal B-splines with control points constrained to lie on the manifold, followed by a closest-point projection onto the man- ifold. Under certain conditions we can prove that these spaces realize the optimal approximation rate. Applications for denoising of manifold-valued data and the com- putation of geometric PDEs will be discussed. This is joint work with Markus Sprecher (ETH Zurich).

INTRINSIC DISCRETIZATION ERROR BOUNDS FOR GEODESIC FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC MINIMIZATION PROBLEMS Hanne Hardering

Freie Universit¨atBerlin, Institut f¨urMathematik, Arnimallee 6, 14195 Berlin, Germany. [email protected]

Geodesic finite elements (GFE) have been introduced recently for energy minimization problems over manifold-valued functions. They are based on the Karcher mean and form a generalization of Lagrangian finite elements. One of the main features of GFE is that they do not rely on an embedding of the manifold into a linear space. We analyze this approximation scheme by presenting a generalization of the standard discretization error bounds for elliptic problems. In particular, we introduce an H1-type error measure and a generalization of the well-known C´ea-Lemma. Combined with the appropriate generalization of interpolation error estimates this yields optimal a priori estimates for H1-elliptic problems, e.g. the approximation of harmonic maps into manifolds with nonpositive or small positive curvature. The results presented are joint work with O. Sander and P. Grohs.

13-4 13: Mini-Symposium: Finite Elements in Nonlinear Spaces

SIMULATION OF Q-TENSOR FIELDS WITH CONSTANT ORIENTATIONAL ORDER PARAMETER IN THE THEORY OF UNIAXIAL NEMATIC LIQUID CRYSTALS Alexander Raisch

Institute for Numerical Simulation, University of Bonn, Bonn, Germany. [email protected]

We propose a practical finite element method for the simulation of uniaxial nematic liquid crystals with a constant order parameter. In the simplest setting this reduces to the task of computing harmonic maps with values in the nonorientable real projective plane. A monotonicity result for Q-tensor fields is derived under the assumption that the underlying triangulation is weakly acute. Using this monotonicity argument we show the stability of a gradient flow type algorithm and prove the converge of outputs to discrete stable configurations as the stopping parameter of the algorithm tends to zero. We examine numerically the difference of orientable and non-orientable stable configurations of liquid crystals in a planar two-dimensional domain and on a curved surface. As an application, we examine tangential line fields on the torus and show that there exist orientable and non-orientable stable states with comparing Landau-de Gennes energy and regions with different tilts of the molecule.

13-5 14: Mini-Symposium: Finite elements for problems with singularities

14 Mini-Symposium: Finite elements for problems with singularities

Organisers: Alexey Bespalov and Serge Nicaise

14-1 14: Mini-Symposium: Finite elements for problems with singularities

EIGENVALUE PROBLEMS IN A NON-LIPSCHITZ DOMAIN Gabriel Acosta and Mar´ıaGabriela Armentanoa

Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, IMAS-Conicet, 1428 Buenos Aires, Argentina. [email protected]

In this work we analyze piecewise linear finite element approximations of the Laplace eigenvalue problem in the plane domain Ω = {(x; y) : 0 < x < 1; 0 < y < xα}; which gives, for α > 1, the simplest model of an external cusp. Since Ω is curved and non- Lipschitz, our problem is not covered by the known literature which, as far as we know, only deals with polygonal or smooth domains. Indeed, the classical spectral theory can not be applied directly and in consequence we present the eigenvalue problem in a proper setting, and relying on known convergence results for the associated source problem with 1 < α < 3 (see [1, 3, 4]), we obtain quasi optimal order of convergence for the eigenpairs [2].

References

[1] G. Acosta and M. G. Armentano (2011), Finite element approximations in a non- Lipschitz domain: Part. II , Math. Comp. 80(276), pp. 1949-1978 .

[2] G. Acosta and M. G. Armentano (2013), Eigenvalue Problem in a non-Lipschitz domain, to appear in IMA Journal of Numerical Analysis.

[3] G. Acosta, M. G. Armentano, R. G. Dur´anand A. L. Lombardi (2005), Nonhomo- geneous Neumann problem for the Poisson equation in domains with an external cusp, Journal of Mathematical Analysis and Applications 310(2), pp. 397-411.

[4] G. Acosta, M. G. Armentano, R. G. Dur´anand A. L. Lombardi (2007), Finite element approximations in a non-Lipschitz domain, SIAM J. Numer. Anal. 45(1), pp. 277-295.

14-2 14: Mini-Symposium: Finite elements for problems with singularities

ANISOTROPIC MESH REFINEMENT IN POLYHEDRAL DOMAINS: ERROR ESTIMATES WITH DATA IN L2(Ω) Thomas Apel1a, Ariel Lombardi2, and Max Winkler1c

1Institut f¨urMathematik und Bauinformatik, Universit¨atder Bundeswehr M¨unchen, Germany [email protected], [email protected] 2Departamento de Matem´atica, Universidad de Buenos Aires, and Instituto de Ciencias, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina [email protected]

The presentation is concerned with the finite element solution of the Poisson equa- tion with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes [2] are used for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)- norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes, see [1].

References

[1] Th. Apel, A. L. Lombardi, and M. Winkler. Anisotropic mesh refinement in poly- hedral domains: error estimates with data in L2(Ω). Submitted, 2013.

[2] Th. Apel and S. Nicaise. The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci., 21:519–549, 1998.

14-3 14: Mini-Symposium: Finite elements for problems with singularities

STRONG CONVERGENCE FOR GAUSS’ LAW WITH EDGE ELEMENTS Patrick Ciarlet1, Haijun Wu2 and Jun Zou3

1 POEMS, ENSTA ParisTech, 828, bd des Mar´echaux, 91762 Palaiseau Cedex, France, [email protected] 2 Department of Mathematics, Nanjing University, Jiangsu, 210093, P.R. China, [email protected], 3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China. [email protected]

We propose and investigate edge element numerical schemes for the time-harmonic Maxwell equations and the stationary Maxwell equations in three dimensions. These approximations have three novel features:

• First, the resulting discrete edge element linear systems can be solved itera- tively with the help of existing preconditioned solvers. Furthermore, numerical experiments show an optimal rate of convergence: the number of iterations is independent of the meshsize.

• Second, no saddle-point discrete systems are required to solve the stationary Maxwell equations.

• Finally, these approximations ensure the strong convergence of the Gauss’ laws in some appropriate norm, in addition to the standard convergence in energy- norm. These estimates hold under general regularity assumptions on the data in (non-convex) polyhedral domains, and for discontinuous coefficients.

In the presentation, we will focus on the norm estimates for the error on the divergence of the fields, and on the new scheme for solving the stationary Maxwell equations. In particular, sharp estimates will be provided for intense fields that may occur in the presence of reentrant corners and/or edges, the so-called geometrical singularities.

14-4 14: Mini-Symposium: Finite elements for problems with singularities

HP -ADAPTIVE FEM BASED ON CONTINUOUS SOBOLEV EMBEDDINGS Thomas Fankhausera, Thomas P. Wihlerb and Marcel Wirzc

Mathematics Institute, University of Bern, Switzerland. [email protected], [email protected] [email protected]

The aim of this talk is to present a new class of smoothness testing strategies in the context of hp-adaptive refinements based on continuous Sobolev embeddings. Here the basic idea in deciding between h- and p-refinement is to monitor the continuity constants of suitable Sobolev inequalities as the hp-FEM spaces are enriched. A few numerical experiments in the context of hp-adaptive FEM for linear elliptic PDE will be performed.

14-5 14: Mini-Symposium: Finite elements for problems with singularities

MAPPING AND REGULARITY RESULTS FOR SCHROEDINGER OPERATORS WITH INVERSE SQUARE POTENTIALS Eugenie Hunsicker1 Hengguang Li2, Victor Nistor3 and Vile Uski4

1Loughborough University, UK [email protected] 2Wayne State, USA 3Penn State, USA 4STFC Harwell Oxford, UK

Consider a Schroedinger operator H with periodic potential in R3. If the potential V is smooth, then the spectrum of H can be studied through Bloch waves associated to vectors k in the first Brillouin zone. The Schroedinger operator Hk associated to such a vector is a smooth elliptic operator on the torus R3/Γ, where Γ is the periodicity lattice of the potential, and thus has a unique self-adjoint extension to the second Sobolev space, H2(R3/Γ). The eigenfunctions of this extension are smooth and form a Hilbert basis of L2(R3/Γ). This means that they can be well approximated using either FEM with arbitrary degree elements or by plane waves. However, if the potential V is of inverse square type, that is V = W/|x − p|2 near a discrete set of points {p} for some continuous function W , and is elsewhere smooth, then we do not automatically get these results about the associated operators Hk. In particular, generally these do not have a unique self-adjoint extension, may not have discrete spectrum and eigenfunctions are not generally smooth. In this talk, I will present some analytic tools which allow us to recover versions of mapping and regularity results for periodic Schroedinger operators with inverse square potentials satisfying the condition that W (p) > −1/4 for all p. V. Nistor will present applications of the results of this talk to FEM for Schrodinger operators with inverse square potentials in his talk.

14-6 14: Mini-Symposium: Finite elements for problems with singularities

FINITE ELEMENT METHOD FOR SCHROEDINGER OPERATORS WITH INVERSE SQUARE POTENTIALS Eugenie Hunsicker1a, Hengguang Li2, Victor Nistor3c, Jorge Sofo3d and Vile Uski1e

1Math. Dept, Loughborough U., Leicestershire, LE11 3TU, UK, [email protected], [email protected] 2Math. Dept, Wayne State U., Detroit, MI 48202, USA, [email protected] 3Math. Dept., Penn State U., University Park, PA 16802, USA, [email protected], [email protected]

We study the regularity and approximability of the eigenvalues and eigenfunctions of periodic Schroedinger operators with inverse square potentials. This is part of a bigger project together with E. Hunsicker, H. Li, J. Sofo, and V. Uski. In my talk, I will discuss the graded mesh approximation of the eigenfunctions, including numerical tests. The results are new even for the usual, Coulomb type potentials (of the form 1/r). We consider, however, the stronger, inverse square potentials (of the form 1/r2) since the difficulties are more pronounced in this case, and thus our method is easier to justify. In particular, the solution can exhibit singularities of the form r−a, with a > 0. In spite of this rather strong singularity, we achieve higher order (hm) optimal rates of convergence.

14-7 14: Mini-Symposium: Finite elements for problems with singularities

LINEAR FINITE ELEMENTS MAY BE ONLY FIRST-ORDER POINTWISE ACCURATE ON ANISOTROPIC TRIANGULATIONS Natalia Kopteva

Mathematics and Statistics Department, University of Limerick, Limerick, Ireland. [email protected]

It appears that there is a perception in the finite-element community that the computed- solution error in the maximum norm is closely related to the corresponding interpola- tion error. While an almost best approximation property of finite-element solutions in the maximum norm has been rigourously proved (with a logarithmic factor in the case of linear elements) for some equations on quasi-uniform meshes, there is no such result for strongly-anisotropic triangulations. Nevertheless, this perception is frequently con- sidered a reasonable heuristic conjecture to be used in the anisotropic mesh adaptation. In this talk, we give a counterexample of an anisotropic triangulation on which • the exact solution is in C∞(Ω)¯ and has a second-order pointwise error of linear interpolation O(N −2), • the computed solution obtained using linear finite elements is only first-order pointwise accurate, i.e. the pointwise error is as large as O(N −1). Here the maximum side length of mesh elements is O(N −1) and the global number of mesh nodes does not exceed O(N 2). Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Both standard and lumped-mass cases are addressed. A theoretical justification of the observed numer- ical phenomena is given by the following lemma (which is established using a finite- difference representation of the considered finite element methods). Suppose Ω ⊃ ˚Ω, where the subdomain ˚Ω := (0, 2ε) × (−H,H) with the tensor- i 2N0 ˚ ˚ product mesh ˚ωh := {xi = ε } ×{−H, 0,H}. The triangulation T in Ω is obtained N0 i=0 by drawing diagonals in each rectangle as shown below, using the mesh transition point

(ε, 0). H

0 T in Ω: T0 in Ω0 ⊂ Ω:

−H 0 ε2 ε

Lemma. Let u = e−x/ε be the exact solution of the equation −ε24u + u = 0, subject to a Dirichlet boundary condition, posed in a domain Ω ⊃ ˚Ω, a triangulation T in Ω be such that T ⊃ T˚, and U be the computed solution obtained using linear finite elements. For any positive constant C2, there exist sufficiently small constants C0 and C1 such −1 that if N0 ≤ C1 and ε ≤ C2H, then −1 max |U − u| ≥ C0N0 . Ω¯

14-8 14: Mini-Symposium: Finite elements for problems with singularities

Similar phenomena will be also discussed in the context of singularly perturbed convection-diffusion equations.

References

[1] N. Kopteva, Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations, Math. Comp. (2013), accepted for publication; http: //www.staff.ul.ie/natalia/pdf/kopteva_mc2012_R1.pdf.

HP FINITE ELEMENT METHODS FOR SINGULARLY PERTURBED TRANSMISSION PROBLEMS Serge Nicaise1 and Christos Xenophontos2

1LAMAV, Universit´ede Valenciennes et du Hainaut Cambr´esis,Francex [email protected] 2Department of Mathematics and Statistics, University of Cyprus [email protected]

We consider singularly perturbed transmission problems with two different diffusion coefficients, in one- and two-dimensions. Their solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high, interface layers along the interface and, in the case of polygons, corner singularities at the vertices of the domain. In each case, we are interested in regularity estimates for each solution component, that are explicit in the differentiation order and the singular perturbation parameter (i.e. the diffusion coefficient). These estimates will guide us in the construction (and analysis) of robust hp finite element methods for the approximation of the solution. Under the assumption of analytic input data, we show that the hp version of the finite element method on so-called Spectral Boundary Layer Meshes yields exponen- tial rates of convergence in the energy norm, as the degree p of the approximating polynomials increases. Numerical results illustrating our theoretical findings will also be presented.

14-9 15: Mini-Symposium: Foundations of isogeometric analysis

15 Mini-Symposium: Foundations of isogeometric analysis

Organisers: Lourenco Beir˜aoda Veiga and Annalisa Buffa

15-1 15: Mini-Symposium: Foundations of isogeometric analysis

TOWARDS ISOGEOMETRIC ANALYSIS FOR COMPRESSIBLE FLOW PROBLEMS AND UNSTRUCTURED MESHES R. Abgrall

INRIA and Universite de Bordeaux, France [email protected]

In this talk, we first show how the residual distribution formalism can be extended to Bezier and Nurbs elements, using unstructured meshes. This leads to parameter free schemes, able to compute possibly discontinous flows. Then we address the meshing issue. Given a CAD, we show how to compute automatically a ”curved” mesh, in two and three dimensions, that is exactly compatible with the CAD representation of the geometry. Using the combination of these tools (scheme and meshing), and degree elevation, we show how to do mesh adaptation. This work was done with C. Dobrzynski and A. Froehly, funding ERC advanced grant ADDECCO # 226632.

15-2 15: Mini-Symposium: Foundations of isogeometric analysis

ARBITRARY-DEGREE ANALYSIS-SUITABLE T-SPLINES L. Beir˜aoda Veiga, Annalisa Buffa, Giancarlo Sangallia and Rafael V´azquez

Dipartimento di Matematica – Universit`adi Pavia, Italy [email protected]

IsoGeometric Analysis (IGA) is a numerical method for solving partial differential equations (PDEs), introduced by Hughes et al. in [3]. In IGA, B-splines or Non- Uniform Rational B-Splines (NURBS), that typically represent the domain geometry in a Computer Aided Design (CAD) parametrization, become the basis for the solution space of variational formulations of PDEs. Local refinement strategies are possible thanks to the non-tensor product extensions of B-splines, such as T-splines ([5, 6]). A T-spline space is spanned by a set of B-spline functions, named T-spline blending functions, that are constructed from a T-mesh. The T-mesh breaks the global tensor- product structure by allowing so called T-junctions. T-splines have been recognized as a promising tool for IGA in [1] and have been the object of recent interest in literature. In particular, in the context of IGA, analysis- suitable (AS) T-splines have emerged: introduced in [4] in the bi-cubic case, they are a sub-class of T-splines for which we have fundamental mathematical properties needed in a PDE solver. Linear independence of AS T-splines blending functions has been first shown in [4]. In [2] it is shown that the condition of being AS, which is mainly a condition on the connectivity of the T-mesh, implies that the bi-cubic T-spline basis functions admit a dual basis that can be constructed as in the tensor-product setting. We present here generalizations of the results of [2] in a fundamental way. We define the the class ASp,q of analysis suitable T-meshes of degrees p, q and show how the abstract theory results in properties of the T-spline space: existence of a dual basis, first of all, and then linear independence of the blending functions and existence of a projector operator with optimal approximation properties.

References

[1] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, and T.W. Sederberg. Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Engrg., 199(5-8):229 – 263, 2010.

[2] L. Beir˜ao da Veiga, A. Buffa, D. Cho, and G. Sangalli. Analysis-Suitable T-splines are Dual-Compatible. To appear in Comput. Methods Appl. Mech. Engrg., 2012.

[3] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135–4195, 2005.

[4] X. Li, J. Zheng, T.W. Sederberg, T.J.R. Hughes, and M.A. Scott. On linear inde- pendence of T-spline blending functions. Comput. Aided Geom. Design, 29(1):63 – 76, 2012.

15-3 15: Mini-Symposium: Foundations of isogeometric analysis

[5] T.W. Sederberg, D.L. Cardon, G.T. Finnigan, N.S. North, J. Zheng, and T. Lyche. T-spline simplication and local refinement. ACM Trans. Graph., 23(3):276–283, 2004.

[6] T.W. Sederberg, J. Zheng, A. Bakenov, and A. Nasri. T-splines and T-NURCCSs. ACM Trans. Graph., 22(3):477–484, 2003.

15-4 15: Mini-Symposium: Foundations of isogeometric analysis

ISOGEOMETRIC ANALYSIS AND NON-MATCHING DOMAIN DECOMPOSITION METHODS Michel Bercovier

Rachel and Selim Benin School of Comp. Sc. and Eng., The Hebrew University of Jerusalem 91904, Jerusalem, Israel.? [email protected]

Isogeometric analysis (IGA) is a rapidly developing paradigm for the discretization of Partial Differential Equations (PDEs). The basic idea detailed in[J. A. Cottrell, T.. J. R.. Hugues, Y. Bazilevs , Isogeometric Analysis ,Wiley,UK, 2009 ] consists in defining the same global isoparametric transformation for the exact computational domain using B-Splines or NURBS and for the approximation functions for the PDEs solution. One of the aims is to avoid the costly steps of mesh generation and CAD interchange. Domain decomposition methods are natural candidates for the solution of large IGA problems and have been studied in [L. Beir˜aoda Veiga, D. Cho, L. F. Pavarino, and S. Scacchi Overlapping Schwarz Methods for Isogeometric Analysis SIAM Journal on Numerical Analysis 2012, Vol. 50, No. 3, pp. 1394-1416], where overlapping domains correspond to matching grids. In order to use some basic constructs of CAD (boolean operations such as union or intersections), we introduced the simplest Schwarz Additive Domain Decomposition Method (SADDM) [O. B. Widlund, A. Toselli, Domain Decomposition Method: Algo- rithms and Theory, Springer, 2004] in IGA. Ωi, i = 1, ..., n , are overlapping ( i.e. there is always a pair (i, j) such that Ωi ∩ Ωj has a non void interior) and that the respective isoparametric transformations are non matching,: the pair of reference grid and knots defining each physical domain are not related. ˆ ˆ The respective inverse isogeometric mapping from Ωi to Ωi defines in Ωj a trimming line ( resp. a trimming surface in 3D), and the corresponding (partial) boundary Γj,i of Ωj In SADDM we need to compute at each iteration the trace of the solution ui obtained in the sub-domain Ωi, on the boundary Γj,i of the sub-domain Ωj. It gives rise to a non homogeneous boundary condition (BC) problem on each sub domain. Application of this condition in IGA is not straightforward. We analyze different approaches such as approximation, least square and others, and compare them. We give several examples illustrating the power of this approach: direct use of CGS primitives, local zooming instead of refinements, and parallelization for large problems. We show that there is no degradation of the powerful approximation properties of IGA when using non matching meshes. The examples are computed by means of a standard open source IGA code for each domain, GeoPDEs [C. de Falco, A. Reali, and R. Vazquez. GeoPDEs: a research tool for Isogeometric Analysis of PDEs. Advances in Software Engineering,40 (2011),1020- 1034.] This search has been done jointly with Ilya Soloveichik, who did most of the exam- ples implementations.

15-5 15: Mini-Symposium: Foundations of isogeometric analysis

IMPLEMENTATION OF HIGH ORDER IMPEDANCE BOUNDARY CONDITIONS IN ISOGEOMETRIC METHODS Annalisa Buffa1a, Luca Di Rienzo2 and Rafael V´azquez1c

1 Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes” del CNR, via Ferrata 1, I-27100, Pavia, Italy [email protected], [email protected] 2 Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, I-20133, Milan, Italy [email protected]

The concept of surface impedance boundary conditions (SIBC) is now well-known in computational electromagnetics. The idea is to replace the equation of the model inside any conductor by approximate boundary conditions on its surface, restricting the computational domain to the exterior of the conductors. The research on SIBC was started by Leontovich in the 40’s. He proposed an SIBC which only takes into account the local tangent plane at each point on the surface. Later, Rytov proposed an extension of Leontovich’s condition based on an asymptotic expansion. The high order SIBCs proposed by Rytov, which are valid for smooth domains, take into account the curvature of the surface and the second tangential derivative of the field. In this work we present the formulation of a scalar two-dimensional problem with high-order SIBCs, and its discretization with isogeometric methods. There are two main advantages of isogeometric methods in this context: the exact computation of the curvature, and the possibility to compute the second tangential derivative when needed. The method is applied to the computation of the electromagnetic fields in multiconductor transmission lines, to show its performance.

15-6 15: Mini-Symposium: Foundations of isogeometric analysis

A COMPUTATIONAL COST ANALYSIS OF ISOGEOMETRIC ANALYSIS Nathan Collier1, Lisandro Dalcin2, David Pardo3, Maciej Paszynski4 and Victor Calo5

1King Abdullah University of Science and Technology, Saudi Arabia [email protected] 2Consejo Nacional de Investigaciones Cient´ıficasy T´ecnicas,Argentina [email protected] 3The University of the Basque Country and Ikerbasque, Spain [email protected] 4AGH University of Science and Technology, Poland [email protected] 5King Abdullah University of Science and Technology, Saudi Arabia [email protected]

In this talk we discuss computational aspects of isogeometric analysis. We present some work estimates based on floating point operation counts for the assembly process as well as review solver cost estimates for B-spline-based Galerkin and collocation methods. We show that the matrix assembly process for collocation methods is economical when compared to both standard Galerkin methods as well as those using highly continuous B-splines. This suggests that despite their inferior convergence properties, collocation methods can be an efficient alternative to Galerkin methods. We then conclude by presenting a large amount of data in the form of a motion chart which highlights the trends predicted by our work estimates.

15-7 15: Mini-Symposium: Foundations of isogeometric analysis

MIXED ISOGEOMETRIC COLLOCATION METHODS FOR THE STOKES EQUATIONS John A. Evansa, Dominik Schillingerb, Ren´eHiemstrac and Thomas J.R. Hughesd

ICES, The University of Texas at Austin, Austin, TX, USA. [email protected], [email protected], [email protected] [email protected]

Recently, isogeometric collocation has been introduced as a means of dramatically re- ducing the cost associated with isogeometric Galerkin methods [1, 2, 3]. Collocation is based on the discretization of the strong form of a given partial differential equation at a discrete set of collocation points and can be viewed as a specialized one-point quadra- ture scheme. In this talk, we present two classes of mixed isogeometric collocation methods for incompressible fluid flow, focusing on the Stokes equations as a simpli- fied model problem. Our first class of methods is based on the use of Taylor-Hood isogeometric elements while our second employs isogeometric divergence-conforming B-splines. We discuss stability and conservation properties of the two classes of collo- cation methods, details of implementation, and linear solution schemes, and we present relevant numerical results demonstrating the effectiveness of the methods. We finish by comparing the computational cost of the mixed collocation methods with the cost associated with mixed isogeometric Galerkin and finite element methods.

References

[1] F. Auricchio, L. Beir˜aoda Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli. Isogeo- metric collocation methods. Mathematical Models and Methods in Applied Sciences, 20:2075-2107, 2010.

[2] F. Auricchio, L. Beir˜ao da Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli. Isoge- ometric collocation for elastostatics and explicit dynamics. Computer Methods in Applied Mechanics and Engineering, 249-252:2-14, 2012.

[3] D. Schillinger, J.A. Evans, A. Reali, M.A. Scott, and T.J.R. Hughes. Isogeometric collocation: Cost comparison with Galerkin methods and extension to hierarchical NURBS discretizations. Submitted for publication, 2013.

15-8 15: Mini-Symposium: Foundations of isogeometric analysis

ALGEBRAIC MULTILEVEL PRECONDITIONING IN ISOGEOMETRIC ANALYSIS

Krishan Gahalauta and Satyendra Tomarb

Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A4040 Linz, Austria [email protected], [email protected]

The isogeometric analysis proposed by Hughes et al. in [1], has received great deal of attention in the computational mechanics community. In this talk we shall present algebraic multilevel iteration (AMLI) methods [2, 3] for solving linear system aris- ing from the isogeometric discretization of elliptic boundary value problems. AMLI methods are based upon the hierarchical splitting of the solution space. We present the multilevel structure of B-Splines and NURBS spaces and their corresponding hi- erarchical spaces. The matrix formulation of coarse grid operators and its hierarchical complementary operators will be discussed for varying regularity of B-Spline basis func- tions. For NURBS, we generate these operators from B-Splines and the corresponding weights. We shall discuss the quality of splitting of spaces which is measured by the constant γ in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality. For a fixed p, the constant γ will be analyzed for different regularities of the B-Spline ba- sis functions. AMLI methods when applied in the framework of isogeometric analysis shows h-independent convergence rates. Supporting numerical results for CBS con- stant γ, and convergence factor and iterations count for linear AMLI V -cycles and W -cycle, and for nonlinear AMLI W -cycle are provided. Numerical results also show that these methods exhibit almost p-independent convergence rates. Numerical tests are performed, in two-dimensions on square domain and quarter annulus, and in three- dimensions on thick ring. Moreover, for a uniform mesh on a unit interval, the explicit representation of B-Spline basis functions for a fixed mesh size h is given for p = 2, 3, 4 and for C0 and Cp−1 smoothness. In this work we present the construction of AMLI methods. A rigorous analysis, particularly for higher p and regularity, is still open and will be the subject of our future research.

References

[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg. 194 (2005), 4135–4195.

[2] O. Axelsson, P.S. Vassilevski. Algebraic multilevel preconditioning methods I. Nu- mer. Math., 1989; 56:157–177.

[3] O. Axelsson, P.S. Vassilevski. Algebraic multilevel preconditioning methods II. SIAM J. Numer. Anal., 1990; 27:1569–1590.

15-9 15: Mini-Symposium: Foundations of isogeometric analysis

GUARANTEED AND SHARP A-POSTERIORI ERROR ESTIMATES IN ISOGEOMETRIC ANALYSIS S.K. Kleissa and Satyendra Tomarb

RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria [email protected], [email protected]

The potential and the performance of isogeometric analysis (IGA), introduced in [1], have been well-studied over the last years for applications from many fields, see the monograph [2]. Though not a pre-requisite, most of the studies of IGA are based on non-uniform rational B-splines (NURBS). Since the straightforward implementation of NURBS leads to a tensor-product structure, local mesh refinement methods are subject of active current research. Despite the fact that adaptive mesh refinement is closely linked to the question of reliable a posteriori error estimation, the latter is still in its infancy stage in isogeometric analysis. Functional-type a posteriori error estimates, see the recent monograph [3] and the references therein, which have also been studied for a wide range of problems, provide reliable and efficient error bounds, which are fully computable and do not contain any generic, un-determined constants. In this talk functional-type a posteriori error estimates for isogeometric analysis will be discussed. By exploiting the properties of NURBS, we will present efficient computation of these error estimates. The numerical realization and the quality of the computed error distribution will be addressed. The potential and the limitations of the proposed approach will be illustrated using several computational examples.

References

[1] T.J.R. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135–4195, 2005.

[2] J. Cottrell, T.J.R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Toward Inte- gration of CAD and FEA. Wiley, Chichester, 2009.

[3] S. Repin. A Posteriori Estimates for Partial Differential Equations. Walter de Gruyter, Berlin, Germany, 2008.

15-10 15: Mini-Symposium: Foundations of isogeometric analysis

LOCAL REFINEMENTS IN IGA BASED ON HIERARCHICAL GENERALIZED B-SPLINES Carla Manni1a, Francesca Pelosi1b and Hendrik Speleers2

1Dipartimento di Matematica, Universit`adi Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy, [email protected], [email protected] 2Departement Computerwetenschappen, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium, [email protected]

Tensor-product generalized B-splines can offer an interesting problem–oriented alter- native to NURBS (Non-Uniform Rational B-Splines) in IgA (Isogeometric Analysis) as investigated in [Manni, C., Pelosi, F., Sampoli, M.L.: Generalized B-splines as a tool in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 200 (2011) pp. 867–881] and [Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis in advection-diffusion problems: tension splines approximation. J. Comput. Appl. Math. 236 (2011) pp. 511–528]. Generalized B-splines are piecewise functions with sections in more general spaces than algebraic polynomial spaces (like classical B-splines). Suitable selections of such spaces – typically including trigonometric or hyperbolic functions – allow an exact rep- resentation of polynomial curves, conic sections, helices and other profiles of salient interest in applications. Moreover, generalized B-splines possess all fundamental prop- erties of algebraic B-splines (recurrence relation, compact minimum support, local linear independence, . . . ) which are shared by NURBS as well. Finally, contrarily to NURBS, they behave completely similar to B-splines with respect to differentiation and integration. Adaptive local refinement is a crucial ingredient in numerical treatment of partial differential equations. However, NURBS rely on a tensor-product structure so they do not allow adequate local refinements. This motivates the interest in alternative structures for IgA that permit local refinements. Among the others, hierarchical splines have been profitably used in this context. The concept of hierarchical bases can be considered for more general spaces than tensor-product B-splines, see [Giannelli, C., J¨uttler,B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Preprint (2012)]. In particular, a hierarchical structure can be built for tensor- product generalized B-splines, with suitable section spaces, as they suffer from the same drawbacks of NURBS with respect to local refinements. In this talk we present a multilevel representation in terms of a hierarchy of tensor- product generalized B-splines, and we discuss its use in the context of IgA. In this way, we can combine the positive properties of a non-rational model with the possibility of dealing with local refinements.

15-11 15: Mini-Symposium: Foundations of isogeometric analysis

EFFICIENT ASSEMBLY METHOD FOR ISOGEOMETRIC DISCRETIZATIONS Angelos Mantzaflaris1 and Bert J¨uttler2

1RICAM, Austrian Academy of Sciences, Linz, Austria [email protected] 2AG, Johannes Kepler University, Linz, Austria [email protected]

In a typical isogeometric analysis (IGA) pipeline we are given a parameterized geometry (physical domain) and a boundary value problem, whose unknown solution field is projected onto a finite-dimensional sub-space, i.e. we restrict ourselves to finding a solution in that space. Then a linear system is generated, consisting of a computed matrix with e.g. mass, stiffness terms, as well as a load vector containing the moments with respect to the right-hand side. The solution of the resulting linear system yields the coefficients of the unknown field in the chosen discretization space. At each of these steps, errors are introduced and accumulate in the final solution. In most cases the principal error sources during the process are the discretization error coming from projection of the solution and the integration error made in the generation step. Even though IGA has a clear advantage regarding the number of degrees of freedom, matrix generation (by means of numerical integration) constitutes a bottleneck in the overall running times of isogeometric simulations. Similarly to finite element analysis, the standard choice for integral evaluation in IGA is Gaussian quadrature. A serious problem of the latter is that nodes-per-element needed increase rapidly with the degree of the B-Spline basis and the dimension of the problem. Recent developments propose specialized quadrature rules for B-spline bases, that reduce the number of quadrature points and weights used. We propose a new, quadrature-free approach, based on interpolation of the “geome- try factor” and fast look-up operations for values of B-spline integrals for the assembly of common matrix operators, such as the mass or stiffness matrices. The geometry factor refers to the contributions of the (Jacobian of) the geometry mapping to the integral transformation (and possibly to contributions of non-constant coefficients), which is the actual non-polynomial part of the integrand. A sufficiently accurate ap- proximation of this factor in terms of B-splines projects the integrand into a piecewise polynomial space, where exact integration is possible, notably by the use of lookup tables. Theoretical error estimates support our experimental results regarding the ef- ficiency and overall convergence rate which are obtained by applying our method to elliptic problems.

15-12 15: Mini-Symposium: Foundations of isogeometric analysis

COMPARISON OF BOUNDARY ELEMENT METHOD DISCRETISATION TECHNOLOGIES FOR ACOUSTIC ANALYSIS Robert N. Simpson1a, Michael A. Scott2, Matthias Taus3, Derek C. Thomas4 and Haojie Lian1

1School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, UK. [email protected] 2Department of Civil and Environmental Engineering, Brigham Young University, Provo, Utah 84602, USA. 3Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA. 4Department of Physics & Astronomy, Brigham Young University, Provo, Utah 84602, USA.

A significant portion of boundary element method implementations make use of La- grangian discretisations to approximate both the geometry and unknown fields where, in the majority of cases, appropriate ’meshing’ software is required to generate suit- able analysis models. This methodology is well-established, but certain deficiencies are known to exist. Perhaps the most significant of these is the disparity between Computer Aided Design (CAD) and analysis models which incurs significant overheads during the prototype design stages, further exaggerated by the iterative nature of design. In ad- dition, the geometrical error seen in Lagrangian discretisations can lead to significant errors in the resulting solution unless extremely fine meshes are used. A recent developing trend is the use of CAD discretisations to approximate both the geometry and unknown fields for analysis. Termed ‘isogeometric analysis’ by Hughes et al.[1], the concept has received great attention, particular in the context of finite element methods where the use of CAD discretisations reveals many attractive prop- erties. In addition, recent attention has focussed on the use of CAD discretisations in boundary element methods [2] which represents a particularly compelling approach where the use of the same model for both CAD and analysis completely overcomes the mesh generation process. In this work we illustrate the fundamental differences between conventional dis- cretisation technology and that used in the isogeometric concept for boundary element acoustic analysis. The importance of geometrical accuracy is investigated where it is found that the isogeometric approach offers significant improvements in accuracy over its Lagrangian counterpart. The ability of the isogeometric approach to provide a truly integrated design and analysis technology is also demonstrated, offering significant ben- efits for practical engineering design.

15-13 15: Mini-Symposium: Foundations of isogeometric analysis

References

[1] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005.

[2] M. A. Scott, R. N. Simpson, J. A. Evans, S. Lipton, S. P. A. Bordas, T. J. R. Hughes, and T. W. Sederberg. Isogeometric boundary element analysis using unstructured T-splines. Computer Methods in Applied Mechanics and Engineering, 254:197–221, 2013.

SPLINES ON TRIANGULATIONS IN ISOGEOMETRIC ANALYSIS Hendrik Speleers

Departement Computerwetenschappen, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium, [email protected]

Isogeometric Analysis (IgA) is a novel paradigm for numerical simulation which com- bines Finite Element Analysis (FEA) with Computer Aided Design (CAD) methods. The CAD representations – usually in terms of tensor-product B-splines or Non- Uniform Rational B-Splines (NURBS) – are used both to describe the geometry and to approximate the unknown solutions of differential equations. Adaptive local mesh refinement is an important ingredient for obtaining efficiently an accurate solution of differential problems. In the context of classical FEA, local mesh refinement strategies are a well established procedure. Unfortunately, the tensor- product structure of NURBS spaces precludes strictly localized refinements. This mo- tivates the interest in alternative structures for IgA that permit local refinements. In this talk we discuss the use of splines on triangulations for the numerical solution of differential equations in the context of IgA. In particular, we focus on Powell-Sabin (PS) splines which are defined on triangulations with a particular macro-structure. These splines can be represented with basis functions possessing similar properties to the classical (tensor-product) B-splines. The PS B-splines form a convex partition of unity, and the coefficients of this representation have a clear geometric meaning. One can also easily define a rational extension of PS splines, so-called NURPS (Non- Uniform Rational PS). NURPS surfaces allow an exact representation of quadrics, and their shape can be locally controlled by control points and weights in a geometrically intuitive way. Thanks to their structure based on triangulations, PS/NURPS splines offer the flexibility of classical finite elements with respect to local mesh refinements. Moreover, they share with standard tensor-product NURBS the increased smoothness, the B- spline-like basis, and the ability to exactly represent profiles of interest in engineering applications as conic sections. Therefore, they constitute a natural bridge between classical FEA and NURBS-based IgA. We will illustrate the use of PS/NURPS splines in IgA with several numerical examples.

15-14 15: Mini-Symposium: Foundations of isogeometric analysis

APPROXIMATION PROPERTIES OF SINGULAR PARAMETRIZATIONS IN ISOGEOMETRIC ANALYSIS Thomas Takacsa and Bert J¨uttlerb

Institute of Applied Geometry, Johannes Kepler University, Linz, Austria [email protected], [email protected]

Isogeometric analysis is a numerical method based on the NURBS representation of CAD models. The geometry mapping that parametrizes a 2-dimensional physical do- main posesses the tensor product structure of bivariate NURBS. Hence the domain is structurally equivalent to a rectangle or to a hexahedron. The special case of singularly parametrized NURBS surfaces is used to represent non-quadrangular domains without splitting. We analyze the approximation properties of the isogeometric test function spaces on singular parametrizations. We present local refinement strategies that lead to ge- ometrically regular splittings of singular patches. Using this we develop a general framework to prove approximation results for singularly parametrized domains in iso- geometric analysis. We prove bounds for the L2 and H1 approximation error for two classes of singular parametrizations of two dimensional domains.

ADAPTIVE HIERARCHICAL B-SPLINES FOR LOCAL REFINEMENT IN ISOGEOMETRIC ANALYSIS Anh-Vu Vuonga and Bernd Simeonb

Felix-Klein-Centre for Mathematics, University of Kaiserslautern, Germany. [email protected] [email protected]

Adaptive simulation is one of the great challenges in Isogeometric Analysis, which com- bines finite elements with techniques from computer aided geometric design (CAGD). Especially, local refinement is a major obstacle because a straightforward approach by using the standard CAGD routines is prevented by the tensor-product structure, which causes the insertion of various superfluous control points. The main topic of this talk is a refinement technique based on hierarchical B-Splines and its variants. It is very flexible and does not suffer under restrictions on degree and continuity and has the advantage of offering local refinement by construction. Furthermore, properties like linear independence are ensured right from the beginning and the hierarchical spline spaces are fully integrated into the isogeometric setting by adopting well-established finite element techniques into this new context. For example, combined with an a posteriori multi-level error estimator this results in a promising adaptive simulation. The talk will introduce the fundamental idea of this approach, discuss its different variants and illustrate it by some numerical examples.

15-15 16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

16 Mini-Symposium: Global and local error esti- mates for problems with singularities or low reg- ularity

Organisers: Alan Demlow and Dmitriy Leykekhman

16-1 16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

ERROR ANALYSIS OF DISCONTINUOUS GALERKIN METHODS FOR THE STOKES PROBLEM UNDER MINIMAL REGULARITY Santiago Badia1, Ramon Codina1, Thirupathi Gudi2 and Johnny Guzm´an3

1Universtitat Politecnica de Catalunya (UPC), Jordi Girona 1-3, 08034 Barcelona, Spain 2Department of Mathematics, Indian Institute of Science Bangalore, 560012 India [email protected] 3Division of Applied Mathematics, Brown University, Providence RI 02912, USA

We analyze several discontinuous Galerkin methods (DG) for the Stokes problem under the minimal regularity on the solution. We assume that the velocity u belongs to 1 d 2 [H0 (Ω)] and the pressure p ∈ L0(Ω). First, we analyze standard DG methods assuming that the right hand side f belongs to [H−1(Ω) ∩ L1(Ω)]d. A DG method that is well defined for f belonging to [H−1(Ω)]d is then investigated. The methods under study include stabilized DG methods using equal order spaces and inf-sup stable ones where the pressure space is one polynomial degree less than the velocity space.

OPTIMALITY OF AN ADAPTIVE FEM FOR CONTROLLING LOCAL ENERGY ERRORS Alan Demlow

Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506–0027 USA [email protected]

A posteriori error estimates and corresponding adaptive FEM for controlling finite element errors only on local subdomains of the overall computational domain have appeared in the context of “two-grid” and parallel adaptive algorithms. In this talk we will present optimality results for an AFEM designed to control local energy er- rors. Issues that arise include the necessity of controlling “pollution” effects of global solution properties on local solution quality and the effects of singularities on adaptive convergence rates.

16-2 16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

OPTIMAL ERROR ESTIMATES FOR THE PARABOLIC PROBLEM IN L∞(Ω; L2([0,T ])) NORM Dmitriy Leykekhman1 and Boris Vexler2

1Department of Mathematics, University of Connecticut, USA. [email protected], 2Technische Universit¨atM¨unchen, Faculty of Mathematics, Boltzmannstraße 3, 85748 Garching b. Munich, Germany [email protected]

In this talk we discuss the local and global error estimates in L∞(Ω; L2([0,T ])) norm for the second order parabolic problem

ut(t, x) − ∆u(t, x) = f(t, x), (t, x) ∈ I × Ω, u(t, x) = 0, (t, x) ∈ I × ∂Ω, u(0, x) = 0, x ∈ Ω, where Ω is a bounded domain in RN , N ≥ 2. The norm is rather non-standard and not usually considered in the finite element literature, however such error estimates are im- portant for example for optimal control problems with point controls. For the N = 2, optimal error estimates were obtained in [D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control, 2013], however such optimal error estimates in higher dimensions pose significant dif- ficulties. In the talk I will discuss the difficulties and discuss a method to obtain such estimates. The method is in the spirit of [A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, 1977] and requires new type of local energy estimates.

16-3 16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

A POSTERIORI ESTIMATION OF HIERARCHICAL TYPE FOR A SCHRODINGER¨ OPERATOR WITH INVERSE SQUARE POTENTIAL Hengguang Li1 and Jeffrey S Ovall2

1Department of Mathematics, Wayne State University, Detroit, MI, USA [email protected] 2Department of Mathematics, University of Kentucky, Lexington, KY, USA [email protected]

We develop an a posteriori error estimate for mixed boundary value problems of the form (−∆ + δ2|x|−2)u = f, for some constant δ > 0, in Ω ⊂ R2, where Ω contains the origin. Here r = |x|. Operators of this sort can arise in applications in quantum mechanics, and require analysis in weighted Sobolev spaces for well-posedness and regularity results, as well as for the development of effective numerical algorithms. We note that the term |x|−2u is of the “same differential order” order as ∆u, so problems of this sort cannot be view as lower-order perturbations of standard elliptic problems. If the origin is in the interior of Ω, u will generically have an |x|δ type singularity at the origin, in addition to the usual boundary singularities at re-entrant corners in the domain and points where the type of boundary condition changes. Therefore, some type of adapted approximation is needed. In two-dimensions a simple grading strategy may be chosen a priori which guar- antees optimal order convergence. Even in this case, however, a cheaply-computable error estimate is desirable, if for no other reason than to establish a practical stop- ping criterion. We present an a posteriori error estimate of hierarchical-type, and argue that it is equivalent to the actual error in energy norm on a family of geometri- cally graded meshes appropriate for singular solutions of such problems. Experiments demonstrate the behavior of the method in practice. Because such meshes can have strong anisotropy, negatively affecting the conditioning of the linear systems, we also present comparisons with adaptively refined meshes driven by local indicators associ- ated with our approach.

16-4 16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

LOCALIZED POINTWISE ESTIMATES FOR THE FULLY NONLINEAR MONGE-AMPERE` EQUATION Michael Neilan

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA [email protected]

In this talk, I will discuss localized pointwise error estimates for finite element meth- ods of the fully nonlinear Monge-Amp`ereequation. In our approach, we treat the fully nonlinear problem as a perturbed linear problem and resort to arguments given by Schatz who studied pointwise estimates of solutions satisfying a perturbed Galerkin orthogonality condition. Numerical examples will be presented which confirm the the- ory.

16-5 16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

ROBUST LOCALIZATION OF THE BEST ERROR WITH FINITE ELEMENTS IN THE REACTION-DIFFUSION NORM Francesca Tantardini1a, Andreas Veeser1b and R¨udigerVerf¨urth2c

1Dipartimento di Matematica, Universit`adegli Studi di Milano, Milano, Italy [email protected], [email protected] 2Fakult¨atf¨urMathematik, Ruhr-Universit¨at,Bochum, DE [email protected]

1 We consider the problem of approximating a function in H0 in the reaction-diffusion 2 2 2 norm |||·||| := k·kL2 +εk∇·kL2 with continuous finite elements on a given triangulation. We prove that the squared global best error is equivalent to the sum of the squares of the local best errors on pairs of elements:

2 X 2 inf |||u − v|||Ω ≤ C inf |||u − P |||ω(E), (1) v∈S0,` P ∈S0,`| E ω(E) where the constant C is independent of ε. The sum on the right-hand side is over all the faces E of the triangulation and ω(E) is the pair of elements sharing the face E. A result of this type has been proved for the H1-seminorm by A. Veeser. There the local errors are on the single elements, and do not involve the coupling between the elements. We show that, for the reaction-diffusion norm, taking the local errors on single elements in place of the pairs entails that the constant in (1) grows with ε−1. Robustness requires to enlarge the local domains, so that the local errors involve the continuity constraint. Our result shows that it is enough to take pairs of adjacent elements. Moreover we prove a variant of (1), where the local errors are defined on pairs that are “minimal” in the context of bisection. This allows to define, for every element, a local indicator that does not depend on the current triangulation, and so is suitable for the tree approximation of P. Binev and R. DeVore. We can thus compute robust near-best approximations, which may be used as benchmarks for other approximations, e.g. Galerkin solutions.

16-6 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions 17 Mini-Symposium: High order finite element meth- ods: A mini symposium celebrating Leszek Demkow- icz’s contributions

Organisers: Jay Gopalakrishnan and Joachim Sch¨oberl

17-1 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

FEM WITH DISCRETE TRANSPARENT BOUNDARY CONDITIONS FOR THE CAUCHY PROBLEM FOR THE SCHRODINGER¨ EQUATION ON THE WHOLE AXIS Alexander Zlotnik

Department of Higher Mathematics at Faculty of Economics, National Research University Higher School of Economics University, Moscow, Russia. [email protected]

The linear time-dependent Schr¨odingerequation is important in various fields of physics, and often it has to be solved in domains unbounded in space. To this end, a number of approaches were suggested, and the technique of discrete transparent boundary condi- tions (TBCs) is known among the best due to complete absence of spurious reflections from artificial boundaries, stable computations and clear mathematical background leading to a rigorous stability theory [1, 2]. But constructing the discrete TBCs for higher order methods is not a simple matter. We consider the Cauchy problem for the 1D Schr¨odingerequation with variable coefficients on the whole axis. To solve it, we study FEM of any order n ≥ 1 on a space segment and of the Crank-Nicolson type in time coupled to the discrete TBCs. We prove stability bounds in L2 and in the energy-like norm uniformly in time, both with respect to initial data and a free term in the defining integral identity. We also study the similar auxiliary method using an infinite mesh on the whole axis. Next, a model FEM for an auxiliary 2nd order ODE with a complex parameter on the half-axis is solved analytically. Studying the matrix pencil related to the stiffness and mass matrices of the reference element and also applying the reproducing series technique, we construct the discrete TBCs in the form of the Dirichlet-to-Neumann map involving a discrete convolution in time. Its kernel is defined in turn as a n-multiple discrete convolution of sequences expressed in terms of the Legendre polynomials. The discrete TBCs allows to restrict exactly the solution of the auxiliary FEM on the whole axis to a finite segment. We also prove a collection of error bounds up to the order O(τ 2+hn+1) in dependence with smoothness of the initial function. We present computational results for such typical problems as the free evolution of the Gaussian wave package and the tunnel effect for rectangular barriers. They clearly show that higher order finite elements coupled to the discrete TBCs are very effective even in the case of strongly oscillating moving solutions and discontinuous potentials. The results are got jointly with I. Zlotnik (MPEI, Russia) and given in part in [3].

References

[1] X. Antoine, A. Arnold, C. Besse et al., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schr¨odingerequations, Commun. Comp. Phys. (2008) 4 729-796.

[2] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with ap-

17-2 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

proximate transparent boundary conditions for the Schr¨odinger equation. Parts I, II, Commun. Math. Sci. (2006) 4 741-766; (2007) 5 267-298.

[3] A. Zlotnik and I. Zlotnik, Finite element method with discrete transparent bound- ary conditions for the time-dependent 1D Schr¨odingerequation, Kinetic and Re- lated Models (2012) 5 639-667.

HIGH ORDER FEM FOR WAVE PROPAGATION: LIKE IT OR LUMP IT Mark Ainsworth

Division of Applied Mathematics, Brown University, 182 George Street, Providence RI., USA Mark [email protected]

High order finite element methods have long been used for computational wave prop- agation for both first order and second order equations alike. A key issue with com- putational wave propagation is the phase accuracy of the methods: getting the waves to propagate with the right speed is often at least as, if not more, important than the convergence rate. The main variants are the finite element method (FEM) and spectral element method (SEM) with each technique having its band of disciples. SEM can be viewed as a higher order mass-lumped FEM. Despite the predominance of these meth- ods, there is comparatively little by way of hard analysis on what each variant offers in terms of phase accuracy, with the result that there is considerable misinformation and confusion in the literature on this topic. In this presentation, we shall attempt to shed some light on the matter, and also briefly mention new methods that improve on both FEM and SEM.

17-3 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

COMMUTING QUASI INTERPOLANTS FOR T-SPLINE SPACES Annalisa Buffa1a, Giancarlo Sangalli2 and Rafael V´azquez1

1 IMATI CNR “E. Magenes”, Via Ferrata 1, 27100 Pavia, Italy a [email protected] 2 Universit`adegli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy

In the papers [1] and [2], the spline discretization of differential forms is proposed and analysed. In particular commuting projectors are constructed by exploiting the tensor product structure of the mesh and the basis functions. This construction can not in principle be extended to meshes with hanging nodes, i.e., to T-meshes and T-splines. In this talk, I will present quasi-interpolant operators for Analysis Suitable T-splines spaces which commute with the exterior derivative and which enjoy L2 stability proper- ties. Thanks to these commuting operators, we can use the theory of the finite element exterior calculus to provide the mathematical analysis of the T-splines discretization of several problems as e.g., Maxwell equations or Darcy flows.

References

[1] A. Buffa, J. Rivas, G. Sangalli, R. Vazquez, Isogeometric Discrete Differ- ential Forms in Three Dimensions, SIAM J. Numer. Anal. 49 (2011), pp. 818-842.

[2] A. Buffa, G. Sangalli, R. Vazquez, Isogeometric analysis in electromagnet- ics: B-splines approximation , Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 17-20, pp. 1143–1152.

17-4 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

A PDE-CONSTRAINED OPTIMIZATION APPROACH TO THE DISCONTINUOUS PETROV-GALERKIN METHOD WITH A TRUST REGION INEXACT NEWTON-CG SOLVER Tan Bui-Thanha and Omar Ghattasb

Institute for Computational Engineering and Sciences, the University of Texas at Austin, USA. [email protected], [email protected]

We introduce a PDE-constrained optimization approach to the discontinuous Petrov- Galerkin (DPG) method. This point of view allows us to use the full force of the state- of-the-art PDE-constrained optimization technique. The details of the our approach will be presented. In particular, we will show how to compute the gradient and Hessian- vector product using efficient adjoint techniques. The gradient and Hessian-vector product are in turn utilized in a trust region inexact Newton-CG to solve for the DPG solution. The advantage of our approach is that it is valid for both linear and nonlinear PDEs. Moreover, our method is guaranteed to converge to the unique solution for well- posed linear PDEs and to a solution for nonlinear PDEs. Numerical results for several PDEs including Laplace, Helmholtz, nonlinear viscous Burger, and compressible Euler equations are presented to justify our new approach.

17-5 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

A POSTERIORI ERROR CONTROL FOR DPG METHODS Carsten Carstensen1, Leszek Demkowicz2 and Jay Gopalakrishnan3

1Institut f¨urMathematik, Humboldt Universit¨atzu Berlin, Unter den Linden 6, 10099 Berlin, Germany. [email protected] 2Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA. [email protected] 3Department of Mathematics & Statistics, PO Box 751, Portland State University, Portland, OR 97207-0751, USA. [email protected]

Discontinuous Petrov Galerkin (DPG) methods was first presented to the community by Leszek Demkowicz in the Babuˇska lecture of MAFELAP 2009. We give a brief overview of this relatively new class of methods and report on the progress in our research into its theoretical properties. DPG methods minimize a residual in a nonstandard dual norm. The method com- bines least-square ideas with hybridization allowing certain dual norms of the residual to be locally computed. An interesting feature of the method is that it comes with a built-in error estimator thus making it attractive for practical use in scenarios requiring adaptivity. The focus of this talk is on the proof of reliability and efficiency of the error estima- tor. A general a posteriori error analysis using the natural norms of the DPG schemes is now available. We show that a locally and inexactly computed residual norm is both a lower and an upper error bound, up to certain data approximation errors. This en- ables abstract and precise computable upper and lower error bounds. We apply these ideas to DPG discretizations of the equations of Laplace, Lam´eand Stokes.

17-6 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

IMPROVED STABILITY ESTIMATES FOR THE HP -RAVIART-THOMAS PROJECTION OPERATOR ON QUADRILATERALS Alexey Chernov1 and Herbert Egger2

1Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Germany, [email protected] 2Department of Mathematics, TU Darmstadt, Germany [email protected]

Stability the hp-Raviart-Thomas projection operator has been addressed e.g. in [D. Schtzau, C. Schwab, and A. Toselli. Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6):2171–2194, 2002]. These results are suboptimal w.r.t. the polynomial degree p. In this talk we present improved stability estimates for the hp- Raviart-Thomas projection operator on quadrilaterals. Such estimates may be useful per-se, but also have important applications, e.g., in the inf-sup stability proofs and a-posteriori error estimation for DG methods.

17-7 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

DPG METHOD FOR WAVE PROPAGATION PROBLEMS, A BETTER UNDERSTANDING Leszek Demkowicz1, Jay Gopalakrishnan2, Jens Markus Melenk3, Ignacio Muga4 and David Pardo5

1ICES, U Texas at Austin, USA [email protected] 2Dept. of Math., Portland State U, USA [email protected] 3IASC, Vienna UT, Austria, [email protected] 4Dept. of Math., Pontifica U Catolica de Valparaiso, Chile, [email protected] 5Dept. of Applied Math., U Basque Country, Spain, [email protected]

Under appropriate assumptions on the domain and boundary conditions, the opera- tor corresponding to the linear acoustics equations is bounded below with a constant independent of wave number k. This, in turn, implies the uniform stabilityresult for the DPG ultraweak variational formulation and the uniform convergence result for the DPG method [1]: the Finite Element (FE) error is bounded by the Best Approxima- tion (BA) error times a k-independent stability constant. The result, unfortunately, does not explain the pollution-free behavior of the DPG method observed in com- putations. Indeed, the DPG solution consists of “fields” (pressure and velocity) and “traces” (trace of pressure and trace of normal velocity). Whereas fields are measured in the pollution-free L2-norm, the traces are measured in a minimum-energy extension norm that hides derivatives and wave number k. As for every hybrid method, DPG represents a “team work”: the FE error (in fields and traces) is bounded by the (com- bined) BA error in fields and traces. And the BA error of traces scales the same way as in the standard FEM: kp+1hp, one power of k too much... We will present a new convergence analysis for the DPG method based on interpret- ing the DPG method as an implicit realization of a Petrov-Galerkin method with con- forming optimal test functions in the sense of Barret and Morton [2]. The globally op- timal test functions are implicitly approximated with weakly-conforming least squares, one might say that the least squares are working backstage for the DPG method. The convergence analys is hinges on Strang’s lemma and, besides the pollution-free L2 BA error, includes a consistency error. The analysis does not prove that the DPG method is pollution-free but it explains better what we observe numerically, and suggests a modification of the trial spaces improving further the stability properties of the method.

17-8 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

References

[1] L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli. “Wavenumber Ex- plicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation”, CMAME, 213-216, pp.126-138, 2012.

[2] J. Chan, J. Gopalakrishnan and L. Demkowicz, “ Global properties of DPG test spaces for convection-diffusion problems”, ICES Report 2013/5.

A SPACE-TIME MULTIGRID METHOD FOR HIGH ORDER TIME DISCRETIZATIONS Martin Gander1, Martin Neum¨uller2a and Olaf Steinbach2b

1Section de Math´ematiques,Universit´eGen`eve, Gen`eve, Switzerland [email protected] 2Institute of Computational Mathematics, Graz University of Technology, Graz, Austria. [email protected], [email protected]

For evolution equations we present a space-time method based on Discontinuous Galerkin finite elements. Space-time methods have advantages when we have to deal with mov- ing domains and if we need to do local refinement in the space-time domain. For this method we present a multigrid approach based on space-time slabs. This method al- lows the use of parallel solution algorithms. In particular it is possible to solve parallel in time and space. Furthermore this multigrid approach leads to a robust method with respect to the polynomial degree which is used for the DG time stepping scheme. Nu- merical examples will be given which show the performance of this space-time multigrid approach.

PARTIAL EXPANSION OF A LIPSCHITZ DOMAIN AND SOME APPLICATIONS Jay Gopalakrishnan1 and Weifeng Qiu2

1Department of Mathematics and Statistics, Portland State University 2Department of Mathematics, City University of Hong Kong [email protected]

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.

17-9 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

DISPERSIVE AND DISSIPATIVE ERRORS IN THE DPG METHOD WITH SCALED NORMS FOR HELMHOLTZ EQUATION Jay Gopalakrishnan1a, Ignacio Muga2 and Nicole Olivares1b

1 Department of Mathematics and Statistics, Portland State University, Portland, OR 97207-0751, USA. [email protected], [email protected] 2 Instituto de Matem´aticas, Pontificia Universidad Cat´olicade Valpara´ıso, Valpara´ıso,Chile. [email protected],

We consider the discontinuous Petrov-Galerkin (DPG) method, where the test space is normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. Studying the application of the method to the Helmholtz equation, we find that better results are obtained, under some circumstances, as the scaling parameter approaches a limiting value. We perform a dispersion analysis on the multiple interacting stencils that form the DPG method. The analysis shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the computed wave approximations. Since the DPG method is a nonstandard least-squares Galerkin method, we compare its performance with a standard least-squares method.

17-10 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

ADAPTIVE AND HYBRIDIZED HERMITE METHODS FOR INITIAL-BOUNDARY VALUE PROBLEMS Thomas Hagstrom1, Daniel Appel¨o2 and Ronald Chen3

1Department of Mathematics, Southern Methodist University, Dallas TX USA [email protected] 2Department of Mathematics and Statistics, University of New Mexico, Albuquerque NM USA; [email protected] 3College of Optical Sciences, Arizona Center for Mathematical Sciences, University of Arizona, Tucson AZ USA; [email protected]

Hermite methods are general-purpose arbitrary-order volume discretizations with a number of attractive properties. Foremost among these is the possibility to inde- pendently evolve large chunks of data within each computational cell over time steps constrained only by domain-of-dependence requirements. Thus high-order Hermite methods maximize the computation-to-communication ratio, which is likely to be of increasing importance to exploit modern multicore architectures. In addition our ex- perience shows that the methods are quite robust - maintaining stability for problems where other methods fail. This talk will focus on two aspects of our ongoing efforts to enhance our Hermite solvers. The first topic is the implementation and analysis of hp-adaptivity in space and time. In space, the p-adaptive implementation allows the use of polynomials of differ- ing degrees in different cells. The only constraint is to maintain local dissipativity in the Hermite interpolation process. We implement h-adaptivity with quadtree/octree spa- tial refinements, using local time-stepping to maintain efficiency. In time we consider the use of adaptive Runge-Kutta methods independently within each cell. The second development concerns the hybridization of Hermite methods with dis- continuous Galerkin (DG) methods to treat problems in complex geometry. The DG methods are used on an unstructured grid layer near physical boundaries and utilize an independent local time step. In practice we have taken this to be as much as an order of magnitude or more smaller than the time step in the Hermite cells. The latter are part of a Cartesian mesh which, ideally, will cover most of the computational volume. Data is interpolated from the DG cells to the Hermite cells as needed, while the Hermite cells provide fluxes to the DG cells. Our experiments show that the inerent dissipativity of the two discretization techniques is sufficient to render the hybrid method stable. Acknowledgement: The work of the first two authors was supported in part by NSF Grant OCI-0905045. The first author was also supported in part by ARO Contract W911NF-09-1-0344. Any conclusions or recommendations expressed in this talk are those of the authors and do not necessarily reflect the views of NSF or ARO.

17-11 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

ON HP -BOUNDARY LAYER SEQUENCES Harri Hakula

Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland [email protected]

The focus of this talk is a practical implementation of Schwab’s concept of hp-approxi- mation of boundary layers [1] in two dimensions. Boundary layer functions are of the form u(x) = exp(−a x/d), 0 < x < L, where d ∈ (0, 1] is a small parameter, a > 0 is a constant. L is the diameter of the domain, in other words, the longest length scale of the problem. Even though the classical p-method, see e.g. [2], is capable of asymptotic superexponential convergence, judicious choice of a minimal number of elements using a priori knowledge of the boundary layers leads to far more efficient solution in the practical range of p. Moreover, in certain classes of problems, it is possible to choose a robust strategy leading to convergence uniform in d. However, the distribution of the mesh nodes depends on p, and over a range of polynomial degrees p = 2,..., 8, say, the mesh is different for every p! In 1D this is relatively simple, but in 2D much more difficult since we must allow for the mesh topology to change over the range of polynomial degrees. For every boundary layer in the problem, one should have an element of width O(p d) in the direction of the decay of the layer. Notice, that if c p d → L as p increases (c constant), the standard p-method can be interpreted as the limiting method. The algorithm discussed here is based on guiding the meshing process by tracking the mesh lines via a geometric data-structure called an arrangement [3]. The geometric information required for moving the mesh lines is handled within the arrangement and the actual meshes are realised using it. However, the meshes need not be topologically equivalent which distinguishes the proposed algorithm from the r-method, where the mesh topology is fixed but the nodes can be moved. The effectiveness of the algorithm is demonstrated using both source and eigenvalue problems in computational mechanics.

References

[1] Ch. Schwab, p- and hp-Finite Element Methods, Oxford University Press, 1998.

[2] B. Szabo and I. Babuˇska, Finite Element Analysis, Wiley, 1991.

[3] H.Hakula, hp-Boundary Layer Mesh Sequences with Applications to Shell Problems, Computers and Mathematics with Applications, 10.1016/j.camwa.2013.03.007, 2013.

17-12 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

DISCONTINUOUS GALERKIN HP -BEM WITH QUASI-UNIFORM MESHES Norbert Heuer1 and Salim Meddahi2

1Facultad de Matem´aticas,Pontificia Universidad Cat´olicade Chile, Santiago, Chile [email protected] 2Departamento de Matem´aticas, Facultad de Ciencias, Universidad de Oviedo, Oviedo, Spain [email protected]

We present and analyze a discontinuous variant of the hp-version of the boundary element Galerkin method with quasi-uniform meshes. The model problem is that of the hypersingular integral operator on an (open or closed) polyhedral surface. We prove a quasi-optimal error estimate and conclude convergence orders which are quasi-optimal for the h-version with arbitrary degree and almost quasi-optimal for the p-version. Numerical results underline the theory. We gratefully acknowledge support by FONDECYT-Chile through project 1110324 and by Ministery of Education of Spain through project MTM2010-18427.

17-13 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

TWO-GRID HP –ADAPTIVE DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS FOR SECOND–ORDER QUASILINEAR ELLIPTIC PDES Paul Houston

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom. [email protected] http://www.maths.nottingham.ac.uk/personal/ph/

In this talk we present an overview of some recent developments concerning the a posteriori error analysis and adaptive mesh design of h– and hp–version discontinu- ous Galerkin finite element methods for the numerical approximation of second–order quasilinear elliptic boundary value problems. In particular, we consider the deriva- tion of computable bounds on the error measured in terms of an appropriate (mesh– dependent) energy norm, as well as for general target functionals of the solution, in the case when a two-grid approximation is employed. In this setting, the fully non- linear problem is first computed on a coarse finite element space VH,P . The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretization on the finer space Vh,p; thereby, only a linear system of equations is solved on the richer space Vh,p. Here, an adaptive hp–refinement algorithm is proposed which automatically selects the local mesh size and local polyno- mial degrees on both the coarse and fine spaces VH,P and Vh,p, respectively. Numerical experiments confirming the reliability and efficiency of the proposed mesh refinement algorithm are presented. This research has been carried out in collaboration with Scott Congreve (University of Nottingham) and Thomas Wihler (University of Bern).

17-14 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

NEW HYBRID DISCONTINUOUS GALERKIN METHODS Youngmok Jeon1 and Eun-Jae Park2

1Department of Mathematics, Ajou University, Suwon, Korea [email protected] 2Department of Mathematics and Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, Korea [email protected]

A new family of hybrid discontinuous Galerkin methods is studied for second-order elliptic equations [3, 4]. Our proposed method is a generalization of CBE method [2] which allows high order polynomial approximations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Our method can be viewed a hybridizable discontinuous Galerkin method [1] using a Baumann-Oden type local solver. First, optimal order error estimates measured in the energy norm are proved for new triangular elements. Numerical examples are presented to show the performance of the method. Next, quadratic and cubic rectangular elements are proposed and optimal order error estimates measured in the energy norm are provided for elliptic equations [4]. Then, this approach is exploited to approximate Stokes equations and Convection-Diffusion equations. Numerical results are presented for various examples.

References

[1] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), pp. 1319–1365.

[2] Y. Jeon and E.-J. Park, Nonconforming cell boundary element methods for elliptic problems on triangular mesh, Appl. Numer. Math. 58 (2008), pp. 800–814.

[3] Y. Jeon and E.-J. Park, A hybrid discontinuous Galerkin method for elliptic prob- lems, SIAM J. Numer. Anal. 48 (2010), no. 5, 1968-1983.

[4] Y. Jeon and E.-J. Park, New locally conservative finite element methods on a rectangular mesh, Numerische Mathematik 123 (2013), 97-119.

17-15 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

GODUNOV SPH METHODS FOR SIMULATING COMPLEX FLOWS WITH FREE SURFACES OVER RAPIDLY CHANGING NATURAL TERRAINS Dinesh Kumar1, E. B. Pitman2b and A. K. Patra2c

1Department of Mech. and Aero. Eng., University at Buffalo, Buffalo, NY 14260, USA [email protected], 2Department of Mathematics, University at Buffalo, Buffalo, NY 14260, USA [email protected], [email protected]

The Lagrangian nature Godunov Smooth Particle Hydrodynamics is naturally suitable for free-surface flows like those modeled as depth averaged (shallow water like model) granular avalanches. Yet enforcing boundary conditions on rapidly changing natural surfaces and high computational cost when compared to its grid-based Eulerian coun- terparts present major challenges. In this talk, we present a three-dimensional imple- mentation of Godunov-SPH method for such flows, on natural terrains. Godunov-SPH is based on the work of Inutsuka [J. Comp.Phys. 2002; 179:238267] that accurately resolves discontinuities without the need to use artificial viscosity. Our approach builds on the basic Inustsuka approach but introduces a modified ghost particle method to correctly enforce the essential boundary conditions and a Navier slip based boundary condition for the natural boundary condition. The development here is motivated by the need to improve upon depth averaged grid based models of large scale debris flows and avalanches, often characterized as granular flows (see the Figure for a sample flow that cannot be simulated using traditional depth average methods).

Sample simulations of a laboratory test of granular flows.

17-16 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

APPLICATION OF HP FINITE ELEMENTS TO THE ACCURATE COMPUTATION OF POLARISATION TENSORS FOR THE EDDY CURRENT PROBLEM P.D. Ledger1 and W.R.B. Lionheart2

1College of Engineering, Swansea University, U.K. [email protected] 2School of Mathematics, The University of Manchester, U.K. [email protected]

Engineers interested in the detection of unexplored ordnance have long since postulated that a formula exists that describes the change in a low frequency magnetic field caused by the presence of a conducting object e.g. [1]. Furthermore, it has been conjectured that the formula contains a polarisation tensor, which describes the shape and material properties of the object. Until very recently, however, it has not been clear how this polarisation tensor can be computed in practice for different objects nor how the change in magnetic field depends on frequency. The key to answering these questions lies in the development of asymptotic expansions of the perturbed magnetic field in appropriate variables. The leading order term in asymptotic expansions of perturbed electromagnetic fields due to the presence of dielectric objects as δ → 0 and r → ∞, where δ is the object size and r the distance from the object to point of observation, has been obtained by Ammari and Volkov[2]. These expansions are expressed in terms of polarisation tensors and hold great potential for the solution of inverse problems where the task is to deter- mine the shape, location and dielectric object properties from far field measurements. Recently, we have obtained the leading order terms in expansions that describe the perturbed fields as max(δ/r, kδ) → 0. Our results are also written in terms of polarisa- tion tensors and describe low frequency perturbed fields at distances large compared to the object size, where k is the free space wave number. It is expected that they will be useful for finding dielectric objects from low frequency near field measurements. In the case of highly conducting objects, at low frequencies, the eddy current approximation of Maxwell’s equations is often made. An important length scale for such problems is the skin depth and, for the case where this is of same order as the object’s size, Am- mari, Chen, Chen et al. [4] have obtained an expansion for the perturbed field magnetic field as δ → 0. This result is expressed in terms of a new conductivity polarisation tensor and a new permeability tensor. The computation of these new tensors requires the solution of a Maxwell transmission problem. These developments offer possibilities for finding conducting objects from low frequency field measurements. In this talk, we shall review these different asymptotic expansions and present an approach for the efficient and accurate computation of the conductivity and polarisation tensors for the eddy current problem using hp edge finite elements.

17-17 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

References

[1] Y. Das, J.E. McFee, J. Toews and G.C. Stuart, IEEE T. Geosci Remote, 28, 278-288, 1990.

[2] H. Ammari and D. Volkov, Int. J. Multiscale Com., 3, 149-160, 2005.

[3] P.D. Ledger and W.R.B. Lionheart, Submitted 2012.

[4] H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov, Submitted 2012.

RECENT ADVANCES IN FINITE ELEMENT SIMULATION OF ELECTROMAGNETIC WAVE PROPAGATION IN METAMATERIALS Jichun Li

Department of Mathematical Sciences, University of Nevada, Las Vegas, USA. [email protected]

Since 2000, there is a growing interest in the study of metamaterials due to their potential applications in areas such as design of invisibility cloak and sub-wavelength imaging, etc. In this talk, I’ll first give a brief introduction to the short history of metamaterials. Then I’ll focus on mathematical modeling of metamaterials, and discuss some finite element schemes we developed in recent years. Finally, I’ll conclude the talk with some interesting simulation results such as backward wave propagation and cloaking simulation. Some open issues will be mentioned too.

References

[1] J. Li and Y. Huang, “Time-Domain Finite Element Methods for Maxwell’s Equa- tions in Metamaterials”, Springer Series in Computational Mathematics, vol.43, Springer, Jan. 2013, 302pp.

[2] L. Demkowicz and J. Li, Numerical simulations of cloaking problems using a DPG method, Computational Mechanics. DOI 10.1007/s00466-012-0744-4

[3] Y. Huang, J. Li and Q. Lin, Superconvergence analysis for time-dependent Maxwell’s equations in metamaterials, Numerical Methods Partial Differ. Equ. 28 (2012) 1794-1816.

[4] J. Li, Finite element study of the Lorentz model in metamaterials, Computer Meth- ods in Applied Mechanics and Engineering 200 (2011) 626-637.

17-18 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

HP -FEM FOR SINGULAR PERTURBATIONS WITH MULTIPLE SCALES Jens Markus Melenk and Christos Xenophontos

Institut f¨urAnalysis und Scientific Computing, Technische Universit¨atWien, Austria Wiedner Hauptstr. 8-10, A-1040 Wien [email protected] Department of Mathematics and Statistics, University of Cyprus PO Box 20537, 1678 Nicosia, Cyprus

We consider the approximation of solutions of systems of elliptic equations that are singularly perturbed. Based on suitably designed meshes, we show that the hp-version of the FEM can achieve robust exponential convergence, i.e., the convergence is uniform with respect to the singular perturbation parameters. The natural energy norm in which this convergence takes place is, however, rather weak and does not “see” the layers. We will report on recent progress for convergence in a stronger norm. This norm features a different scaling of the parameters and is such that the layers are uniformly O(1) as the singular perturbation parameters tend to zero. Thus, convergence in this norm yields relevant information about the convergence within the layer.

HP ADAPTIVE FINITE ELEMENT METHODS BASED ON DERIVATIVES RECOVERY AND SUPERCONVERGENCE Hieu Nguyen1 and Randolph E. Bank2

1 Department of Mathematics, Heriot-Watt University, Edinburgh, UK. [email protected] 2 Department of Mathematics, University of California, San Diego, USA.

In this talk, we present a hp-adaptive finite element method based on derivative recov- ery and superconvergence. In our approach, high-order derivatives of the solution are estimated by superconvergent approximations. These approximations are then used to formulate a posteriori error estimates for guiding adaptivity. The decision whether it is beneficial to refine a given element in h (geometry) or in p (degree) is made via verifying a superconvergence result on that element. A special set of transition basis functions is also introduced to guarantee continuity across elements of different degrees. Numerical results shows that our method achieves exponential rate of convergence predicted by theory.

17-19 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

COMPARISON OF DIFFERENT FINITE ELEMENT MODELS AND METHODS FOR THE GIRKMANN SHELL-RING PROBLEM Antti H. Niemia and Julien Petitb

Department of Civil and Structural Engineering, Aalto University, Espoo, Finland [email protected], [email protected]

Finite element analysis of thin shell structures involves various explicit and implicit modelling assumptions that extend way beyond the limits of mathematical conver- gence theory currently available. However, thanks to modern computation technology, such as the hp-adaptive finite element method as pioneered by Leszek Demkowicz, computations can also be based directly to elasticity theory. This approach rules out the modelling errors arising from the use of dimensionally reduced structural models but requires in general more degrees of freedom. In this talk, we will present computational results for the so called Girkmann bench- mark problem involving a spherical shell stiffened by a foot ring. In particular, we will compare the accuracy of finite element models based on elasticity theory to models based on dimensionally reduced structural models.

PYRAMIDAL FINITE ELEMENTS Nilima Nigama, Argyrios Petrasb and Joel Phillipsc

Department of Mathematics, Simon Fraser University, Burnaby, Canada. [email protected], [email protected], [email protected]

In this talk we present a construction of a family of high-order conforming finite ele- ments for pyramidal elements. This family satisfies the commuting diagram property, ensuring stability of mixed finite element discretizations. The analysis of errors due to quadrature is non-standard for these elements, and we describe the key ideas. Finally, we present some new developments towards a family of serendipity elements on the pyramid. This work was motivated in large part by discussions with Prof. Leszek Demkowicz, whose many contributions to the study of FEM we would like to honour.

17-20 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

APPLICATION OF THE ADAPTIVE FINITE ELEMENT METHOD TO NUMERICAL SIMULATIONS OF ARTERIES Waldemar Rachowicz1 and Adam Zdunek2

1Institute of Computer Science, Cracow University of Technology ul. Warszawska 24, 31-155 Cracow, Poland [email protected] 2Swedish Defence Research Agency, Stockholm, Sweden [email protected]

Computational biomechanics is a rapidly developing branch of computer simulations based on the Finite Element Method approximation. An area of special interest is mechanics of soft tissues, especially arteries with aneurysms, or with changes due to atherosclerosis. Both imply a high risk for rupture when pressurized. A commonly accepted approach to biomechanical simulation is to use low order very dense finite element (FE) meshes to discretize the biological objects of interest. Such an approach is motivated by a necessity to accurately approximate a complex patient specific geometry of a piece of artery. It is well known that one can use relatively coarse FE meshes and adapt them using h-refinements (changing locally the size h of elements) and/or p-refinements (changing locally the order of approximation p). The mesh adaptations are intended to reduce the estimated discretisation error below a prescribed threshold at the minimum computational effort. The question arises if the techniques of error estimation and mesh adaptivity could be of any use for computational biomechanics. In this paper we apply the machinery of error estimation and adaptivity for Fi- nite Element simulations of artery problems. Appropriate algorithms exist to some extent. Some adjustments are however needed. We developed the two-field formula- tion of displacement-pressure type [1] with no inter-element continuity enforced for the pressure variable. We developed an error estimation technique which is applicable to finite elasticity problems. It is a version of the residual method proposed by Bank and Weiser in the context of linear elliptic boundary-value problems. The distribution of error indicators may be used to guide adaptivity of finite element meshes. We applied adaptive FEM to solve typical problems of mechanics of arteries. We intend to use the adaptive version of the FEM to simulate the interaction between a medical device, namely a stent, and the artery. The stent is inserted into the artery and it is adequately deformed to widen the cross-section which has been narrowed due to the atherosclerosis. A precise control of this treatment, called angioplasty, is necessary. This simulation involves a contact problem with large sliding and with finite deformation of the bodies in contact.

17-21 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

References

[1] Simo, J. C. and Taylor, R. L., Quasi-incompressible finite elasticity in principal stretches, Comput. Methods in Appl. Mech. Engng, 85, pp. 273−310, 1991.

DISCONTINUOUS PETROV-GALERKIN METHODS FOR INCOMPRESSIBLE FLOW: STOKES AND NAVIER-STOKES Nathan V. Robertsa, Leszek Demkowiczb and Robert Moserc

Institute of Computational Engineering and Sciences, University of Texas, Austin, TX, USA. [email protected], [email protected], [email protected]

The discontinuous Petrov-Galerkin (DPG) finite element methodology first proposed in 2009 by Demkowicz and Gopalakrishnan [1, 2]—and subsequently developed by many others—offers a fundamental framework for developing robust residual-minimizing fi- nite element methods, even for equations that usually cause problems for standard methods, such as convection-dominated diffusion and the Stokes equations. For a very broad class of well-posed problems, DPG offers provably optimal convergence rates with a modest convergence constant. In some of our experiments, DPG not only achieves the optimal rates, but gets extremely close to the best approximation available in the discrete space. Moreover, DPG provides a way to measure—not merely estimate—the error in the approximate solution, which can then robustly drive adaptivity. In this presentation, we apply DPG to Stokes and Navier-Stokes, discussing theo- retical convergence estimates [3] and illustrating these through numerical experiments performed with Camellia [4], a robust, flexible software framework for DPG research and experimentation, built atop Trilinos.

References

[1] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation. Computer Methods in Applied Mechanics and Engineering, 199(23-24):1558 – 1572, 2010.

[2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov–Galerkin methods. Part II: Optimal test functions. Numerical Methods for Partial Differen- tial Equations, 27(1):70–105, 2011.

[3] Nathan V. Roberts, Tan Bui-Thanh, and Leszek F. Demkowicz. The DPG method for the Stokes problem. Technical Report 12-22, ICES, 2012.

[4] Nathan V. Roberts, Denis Ridzal, Pavel B. Bochev, and Leszek F. Demkowicz. A Toolbox for a Class of Discontinuous Petrov-Galerkin Methods Using Trilinos. Technical Report SAND2011-6678, Sandia National Laboratories, 2011.

17-22 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

PRECONDITIONING FOR HIGH ORDER HYBRID DG METHODS Joachim Sch¨oberl1 and Christoph Lehrenfeld2

1Institute for Analysis and Scientific Computing, Vienna UT, Austria [email protected] 2 Institut f¨urGeometrie und Praktische Mathematik, RWTH Aachen, Germany [email protected]

Discontinuous Galerkin methods provide a lot of freedom for the design of finite element methods, such as upwinding for convection dominated problems, or the choice of stable mixed finite elements. Hybridization of DG methods allows an efficient implementation in terms of matrix entries, and static condensation. In this talk we focus on the analysis of preconditioners for the hybrid-DG method for elliptic problems. We prove a poly-logarithmic condition number in h and p for 2D as well as 3D. Techniques are p-version extension operators from vertices, edges, and faces with respect to the norm induced by the stabilization term. We show that the Bassi-Rebay method differs essentially from the interior penalty method, in 3D. Numerical results for the model problem as well as more complex problems confirm the theoretical results. [J. Sch¨oberl and C. Lehrenfeld: Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes, in Advanced Finite Element Methods and Applications, Lecture Notes in Applied and Computa- tional Mechanics 66, Springer 2012, pages 27-56]

17-23 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

APPLICATION OF THE FULLY AUTOMATIC HP -FEM TO ELASTIC-PLASTIC PROBLEMS Marta Serafina and Witold Cecotb

Cracow University of Technology, Krakow, Poland [email protected], [email protected]

The fully automatic (self automatic) hp-adaptive mesh refinement strategy [1] was applied to analysis of elastic-plastic problems. Since the solution is less regular at the elastic-plastic interface, the finite element meshes should comply with elastic and plastic zones. However, the elastic and plastic zones are not known a-priori thus, appropriate adaptive mesh refinements are the way to construct meshes that, at least, approximately correspond to the shapes of the zones. Generally, the self adaptive mesh refinement technique generates the aforemen- tioned meshes and, for the considered physically nonlinear problems, delivers the fastest convergence of the error. We tested additional h-refinements and p-enrichment along the elastic-plastic interface, as well as exclusively h-adaptive or p-adaptive mesh refine- ments. Only in the case of cylinder additional h-refinements resulted in a speed up of the convergence. Presumably the reason for significant improvement only in this case was the shape of elastic-plastic interface, which could be relatively easily captured in the cylinder problem. In the other examples even anisotropic additional h-refinements did not result in meshes that exactly complied with elastic-plastic zones. Therefore, the original hp-FEM delivered the fastest algebraic convergence for elastic-plastic problems resulting in meshes with up to 8th order of approximation. In the future the fully au- tomatic hp mesh refinements will be supplemented with the r-adaptation, since it was successfully used for p-refinements [2]. Also the algorithm of searching for the optimal new hp meshes should undergo further testing. Currently, for the sake of efficiency, only certain selected from all possible refinements are considered. Such a strategy works correctly for linear problems but its validation for elastoplasticity is intended.

References

[1] L. Demkowicz, W. Rachowicz, and Ph. Devloo. A fully automatic hp-adaptivity. Journal of Scientific Computing, 17:127–155, 2002.

[2] V. N¨ubel and A. D¨usterand E. Rank An rp-adaptive finite element method for the deformation theory of plasticity. Comput. Mech., 39:557–574, 2007.

17-24 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

A NEW ERROR ANALYSIS FOR CRANK-NICOLSON GALERKIN FEMS FOR A GENERALIZED NONLINEAR SCHRODINGER¨ EQUATION Jilu Wang

Department of Mathematics, City University of Hong Kong, Hong Kong [email protected]

This talk is concerned with unconditionally optimal L2 error estimates of linearized Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schr¨odinger equation, while certain conditions on time stepsize were always needed in previous works. A key to our analysis is an error splitting, in terms of the corresponding time-discrete system, with which the error is split into two parts, temporal error and spatial error. We can prove that the spatial error is τ-independent. By the inverse inequality, the boundedness of numerical solution in L∞-norm follows immediately. Then, the optimal L2 error estimates are obtained by a routine method. Numerical results in both two and three dimensional spaces are given to confirm our theoretical analysis.

B-SPLINE FEM APPROXIMATION OF WAVE EQUATION Hongrui Wanga and Mark Ainsworth

Department of Mathematics and Statistics, University of Strathclyde, UK [email protected]

The use of high order splines for the approximation of PDEs has recently attracted a lot of attention due to the work of Hughes [1]. However, it is as yet unclear how these methods perform as the order of approximation is increased on a fixed mesh. We analyse this problem in the setting of the wave equation. We also show how the methods can be implemented efficiently and analyse its stability.

References

[1] John A. Evans, Yuri Bazilevs, Ivo Babuska, and Thomas J.R. Hughes. n-widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. Computer Methods in Applied Mechanics and Engineering, 198:1726 – 1741, 2009.

17-25 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

THE LOW-STORAGE CURVILINEAR DISCONTINUOUS GALERKIN METHOD T. Warburton

Department of Computational and Applied Mathematics, Rice University, Houston, TX, USA. [email protected]

Accurate modeling of wave scattering by curvilinear geometries can be achieved with curvilinear finite element methods. In this talk we will describe a memory efficient discontinuous Galerkin method for scattering from curvilinear objects [1, 2, 3]. We will present an a priori convergence analysis and compare method specific sufficient conditions guaranteeing convergence for both the low-storage and traditional variants of the discontinuous Galerkin method. We will present computational results that confirm the analysis and also demonstrate the potential for high performance computations accelerated by graphics processing units.

References

[1] T. Warburton. A low storage curvilinear discontinuous galerkin time-domain method for electromagnetics. In Electromagnetic Theory (EMTS), 2010 URSI In- ternational Symposium on, pages 996–999. IEEE, 2010.

[2] T. Warburton. Aspects of the a priori convergence analysis for the low storage curvi- linear discontinuous Galerkin method. In Theory and Applications of Discontinuous Galerkin Methods, pages 44–46. Mathematisches Forschungsinstitut Oberwolfach, 2012.

[3] T. Warburton. Analysis of the low-storage curvilinear discontinuous galerkin method for wave problems. SIAM Journal on Scientific Computing, Submitted.

17-26 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

A NOVEL FORMULATION FOR NEARLY INEXTENSIBLE AND NEARLY INCOMPRESSIBLE FINITE HYPERELASTICITY Adam Zdunek1, Waldemar Rachowicz2 and T. Eriksson3

1Swedish Defence Research Agency FOI, SE-164 90 Stockholm, Sweden, [email protected] 2Cracow University of Technology, Pl 31-155 Cracow, Poland, [email protected] 3School of Mathematics and Statistics, University of Glasgow, Glasgow, UK. [email protected]

We present a novel formulation of the Hu-Washizu type suited for the modelling of nearly incompressible materials soft in shear and reinforced by rapidly stiffening or nearly inextensible fibres or cords. Its strong form for a Spencer type [1] fibre-reinforced finite hyperelastic material is derived and illustrated with a couple of simple examples. For a single family of fibres it is a 5-field formulation in terms of displacement, an auxil- iary volume ratio, an auxiliary fibre stretch, and the corresponding Lagrange multipliers representing mean pressure and deviatoric fibre stress, respectively. The formulation generalises the Simo-Taylor three-field formulation for near incompressibility [2], often used today in the field of soft tissue biomechanics [3, Sect. 4]. The right Cauchy-Green stretch tensor in the novel formulation is unimodular and in addition it is stretch free in a fibre direction. Furher, the novel formulation corrects a flaw in the class of very popular so-called standard reinforcing models [4], [3, Eqs. (33)-(36)]. The introduced uni-modular and stretch free right Cauchy-Green tensor corrects the modelling error, i.e. it removes the erroneous strain energy contribution in the fibre direction from the ground sub- stance deformation. Simple analytical examples illustrate the modelling error caused using the standard reinforcing model. The setting for the limiting incompressible and inextensible cases and for the fully coupled unconstrained case are also covered.

References

[1] A. J. M. Spencer, Continuum theory of the mechanics of fibre-reinforced compos- ites, Springer, New York, (1984).

[2] J. C. Simo , R. L. Taylor , K. S. Pister , Variational and projection methods for the volume constraint in finite deformation elasto-plasticity., Computer Methods in Applied Mechanics and Engineering 51 (1985) 177–208.

[3] G.A. Holzapfel Structural and numerical models for the (visco)elastic response of arterial walls with residual stresses. In: G.A. Holzapfel and R.W. Ogden (eds.), ”Biomechanics of Soft Tissue in Cardiovascular Systems”, CISM Courses and Lec- tures No. 441, International Centre for Mechanical Sciences, Springer: Wien, New York (2003), 109–184.

17-27 17: Mini-Symposium: High order finite element methods: A mini symposium celebrating Leszek Demkowicz’s contributions

[4] D. A. Poligone, C. O. Horgan, Cavitation for incompressibile anisotropic nolinearly elastic spheres. J. Elasticity, 33 (1993) 27–65.

17-28 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions 18 Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equations

Organisers: Andrea Cangiani and Gianmarco Manzini

18-1 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

BASIC PRINCIPLES OF VIRTUAL ELEMENT METHODS L. Beir˜aoda Veiga1, Franco Brezzi2, Andrea Cangiani3, Gianmarco Manzini4, L.D. Marini5 and Alessandro Russo6

1 Dipartimento di Matematica, Universit`adi Milano Statale, Italy [email protected] 2 Department of Mathematics, University of Leicester, UK [email protected] 3 IUSS-Pavia and IMATI-CNR, Pavia, Italy and KAU, Jeddah, Saudi Arabia [email protected] 4 Group T-5, MS-B284 Los Alamos National Laboratory Los Alamos, NM 87545, USA [email protected] 5 Dipartimento di Matematica, Universit`adi Pavia and IMATI-CNR, Pavia, Italy [email protected] 6 Department of Mathematics and Applications, University of Milano-Bicocca, Italy [email protected]

The Virtual Element Method (VEM) provides a framework for the extension of classical finite element methods to general meshes. A virtual element space is a generalised finite element space consisting of both polynomials and non polynomials functions. The virtual space and degrees of freedom are carefully chosen so that the assembly of the discrete formulation can be based solely on the degrees of freedom, as in, say, (mimetic) finite difference and finite volume approaches, while the finite element setting permits us to analyse the method in the typical finite element fashion. This setting can easily deal with complicated element geometries and/or higher- order continuity conditions (like C1, C2, etc.).

18-2 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

MIMETIC DISCRETIZATIONS OF ELLIPTIC PROBLEMS Gianmarco Manzini

Group T-5, MS-B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected]

We present the family of mimetic finite difference method and discuss how these ap- proach can be used to design efficient schemes with arbitrary order of accuracy and arbitrary regularity to approximate elliptic problems. The diffusion tensor may be heterogeneous, full and anisotropic. These numerical techniques can be applied to computational meshes of polygonal or polyhedral cells with very general shape, also non-conforming as the ones of the Adaptive Mesh Refinement (AMR) method and non-convex.

18-3 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

A VIRTUAL ELEMENT METHOD WITH HIGH REGULARITY

L. Beir˜aoda Veigaa and Gianmarco Manzini

a Dipartimento di Matematica – Universit`adi Milano, Italy [email protected]

The Virtual Element Method (VEM, introduced in [1]) is a generalization of the Finite Element method that, by avoiding an explicit construction of discrete shape functions, achieves a higher degree of flexibility in terms of meshes and properties of the scheme. In order to neglect the explicit construction of basis functions, the method makes use of an approximated bilinear form ah(·, ·) that mimics the original bilinear form a(·, ·). For well-posedness and convergence to be guaranteed, the discrete bilinear form must satisfy precise conditions of stability and consistency. Nevertheless such conditions leave a lot of additional freedom in the construction of the method. In the present talk this feature allows us to design a family of numerical methods [2] that are associated with discrete spaces with arbitrary Cα regularity, polynomial degree m ≥ α + 1 and are suitable to general unstructured polygonal meshes. The parameter α determines the global smoothness of the underlying discrete space, while the parameter m determines the convergence rate for regular solutions. After introducing and describing the method we will show both a-priori [2] and a-posteriori [3] error estimates for the scheme. In particular, the a-posteriori error estimator is composed of various terms that are associated to the various sources of error in the VEM discretization. We finally show a selection of convergence tests, both for uniform and adaptively generated mesh families.

References

[1] L. Beir˜aoda Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo. Ba- sic principles of Virtual Element Methods. Math. Mod. Meth. Appl. Sci., 23(1):199– 214, 2013.

[2] L. Beir˜ao da Veiga, G. Manzini. A Virtual Element Method with arbitrary regu- larity. LANL technical report and in press on IMA J. Numer. Anal..

[3] L.Beir˜aoda Veiga, G. Manzini. Residual a-posteriori error estimation of a Virtual Element Method. Submitted for publication.

18-4 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

THE VIRTUAL ELEMENT METHOD FOR GENERAL SECOND-ORDER ELLIPTIC OPERATORS ON POLYGONAL AND POLYHEDRAL MESHES Franco Brezzi1, L. Donatella Marini2 and Alessandro Russo3

1IUSS-Pavia and IMATI-CNR, Pavia, Italy and KAU, Jeddah, Saudi Arabia [email protected] 2Department of Mathematics, University of Pavia, Italy and IMATI-CNR, Pavia, Italy [email protected] 3Department of Mathematics and Applications, University of Milano-Bicocca, Italy and IMATI-CNR, Pavia, Italy [email protected]

General polygonal and polyhedral meshes naturally arise in the treatment of complex solution domains and heterogeneous materials (e.g. reservoir models) and are particu- larly suited to moving meshes techniques as well as to adaptive mesh refinement and de-refinement. In this talk we will present the Virtual Element Method and show how it can be used to approximate the solution of an elliptic second-order operator on a polygonal or polyhedral mesh with high-order accuracy.

18-5 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

NONSMOOTH INITIAL DATA ERROR ESTIMATES FOR THE FINITE VOLUME ELEMENT METHOD FOR A PARABOLIC PROBLEM Panagiotis Chatzipantelidis

Department of Mathematics, University of Crete, Heraklion, GR–71003, Greece. [email protected]

In this talk we present the results of [1], where we study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that for smooth initial data, we obtain optimal results of second order as in the finite element method for piecewise linear functions. However, for initial data only in L2, a special condition is required, which is satisfied for sym- metric triangulations. Without such a condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for less restrictive meshes triangulations that are almost symmetric or piecewise almost symmetric.

References

[1] P. Chatzipantelidis, R. D. Lazarov, and V. Thom´ee, Some error estimates for the finite volume element method for a parabolic problem, Computational Methods in Applied Mathematics, ( published online), DOI: [2]10.1515/cmam-2012-0006, Jan- uary 2013.

18-6 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

CONVECTION DOMINATED DISCONTINUOUS GALERKIN MULTISCALE METHOD Daniel Elfversona and Axel M˚alqvistb

Division of Scientific Computing, Uppsala University, Sweden. [email protected], [email protected]

In this work we study the solution of the convection dominated convection-diffusion- reaction problems, with heterogeneous and highly varying coefficients without any assumptions on scale separation or periodicity. Problems o this type arise in many branches of scientific computing. There are two reasons why standard continuous finite element methods perform poorly for this kind of problems. That is, both the coefficients describing the problem as well as boundary layers in the solution needs to be resolved. To this end, we propose a new Convection dominated discontinuous Galerkin multiscale method, based on a corrected basis calculated on localized patches, that takes the variations the fine variations into account without resolving it globally ms on a single mesh. Let uH be the solution obtained by the multiscale method where H is the mesh size. Then the following result holds

ms |||u − uH ||| ≤ |||u − uh||| + CH, under moderate assumptions on the magnitude of the convection and that the size the patches where the corrected basis is computed are chosen as O(H log(H−1)). The constant C is independent of the variation in the coefficients nor the mesh size, and uh is the (one scale) discontinuous Galerkin solution on the fine scale. This result holds independent of the regularity of the solution u.

18-7 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

MHM METHOD FOR ADVECTIVE-REACTIVE DOMINATED MODELS Christopher Hardera, Diego Paredesb and Fr´ed´ericValentinc

Laborat´orioNacional de Computa¸c˜aoCient´ıfica- LNCC, Av. Get´ulio Vargas, 333 - 25651-075 Petr´opolis - RJ, Brazil [email protected], [email protected], [email protected]

This work proposes a new family of Multiscale Hybrid-Mixed (MHM) finite element methods for advective-reactive dominated problems on coarse meshes. The MHM method is a consequence of a hybridization procedure. It results in a method that nat- urally incorporates multiple scales while providing solutions with high-order precision for the primal and dual (or flux) variables. The local problems are embedded in the upscaling procedure and are completely independent, meaning they can be naturally obtained using parallel computation facilities. Also, the MHM method preserves local conservation properties from a simple post-processing of the primal variable. The analysis results in a priori estimates showing optimal convergence in natural norms and provides a face-based a posteriori estimator. Regarding the latter, we prove that reliability and efficiency hold. Also, we introduce a new space adaptive strategy which avoids any topological changes on the mesh. Numerical results verify theoretical results as well as a capacity to accurately incorporate heterogeneous and high-contrast coefficients, and to approximate boundary layers. In particular, we show the great performance of the new a posteriori error estimator in driving space adaptivity. We conclude that the MHM method, along with its associated a posteriori estima- tor, is naturally shaped to be used in parallel computing environments and appears to be a highly competitive option to handle realistic multiscale singular perturbed boundary value problems with precision on coarse meshes.

References

[1] C. Harder, D. Paredes and F. Valentin A Family of Multiscale Hybrid-Mixed Finite Element Methods for the Darcy Equation with Rough Coefficients. LNCC Research Report No. 02/2013. To appear in Journal of Computational Physics.

[2] R. Araya, C. Harder, D. Paredes and F. Valentin Multiscale Hybrid-Mixed Method. LNCC Research Report No. 03/2013.

[3] C. Harder, A.L. Madureira and F. Valentin New Finite Elements for Elasticity in Two and Three-Dimensions. LNCC Research Report No. 04/2013.

18-8 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

TREFFTZ-DG METHODS FOR WAVE PROPAGATION: HP -VERSION AND EXPONENTIAL CONVERGENCE Andrea Moiola1, Ralf Hiptmair2a, Ilaria Perugia3 and Christoph Schwab2b

1Department of Mathematics and Statistics, University of Reading, UK [email protected] 2Seminar for Applied Mathematics, ETH Z¨urich, Switzerland [email protected], [email protected] 3Department of Mathematics, University of Pavia, Italy [email protected]

The phenomena involving propagation and interaction of acoustic, electromagnetic and elastic linear waves in time-harmonic regime are often discretised using finite element methods. However, as soon as the wavelength becomes small compared to the diameter of the domain, simulations become excessively expensive. This is due to the highly os- cillatory structure of the solutions in the high frequency regime and to the accumulation of phase error, called numerical dispersion, which affects any local discretisation. To cope with these fundamental difficulties, several recent methods incorporate in- formation about the equations in the design of the trial space. This can be achieved by using Trefftz methods, i.e. choosing test and trial functions that are piecewise solu- tions of the underlying PDE. Typical choices are plane, circular, spherical and angular waves. Prominent examples of such methods are the ultra weak variational formulation (UWVF) of Cessenat and Despr´es;the discontinuous enrichment method (DEM/DGM) of Farhat and co-workers; the variational theory of complex rays (VTCR) of Ladev´eze; and the wave based method (WBM) of Desmet. We focus on a family of Trefftz-discontinuous Galerkin (TDG) schemes, which in- cludes the UWVF as a special case. In the case of the Helmholtz and Maxwell’s equations, a complete theory for the (a priori) convergence in h and p has been devel- oped, see e.g. [A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonic wave problems, PhD thesis, ETH Z¨urich, 2011] and the references therein. This anal- ysis is made possible by the special DG framework used, which ensures unconditional stability and quasi-optimality (i.e., control of the error for any value of the wavenum- ber and the meshsize), and by the use of new approximation estimates for plane and spherical waves, which ensure high-order convergence. A special choice of the numeri- cal flux parameters allows to prove a priori error estimates in L2-norm for meshes that are locally refined, for example near the corners of a scatterer [R. Hiptmair, A. Moiola, I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes, Appl. Numer. Math., 2013]. However, the exponential convergence in the number of DOFs of a full hp-version of the TDG is a highly desirable result. To achieve this, new fully explicit error bounds for the approximation of har- monic functions by harmonic polynomials in star-shaped elements have been proved in [R. Hiptmair, A. Moiola, I. Perugia, Ch. Schwab, Approximation by harmonic polyno- mials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM, SAM report 2012-38, ETH Z¨urich] relying on complex variable techniques√ and following the argument of M. Melenk. These bounds were used to prove exp(−b #dofs) orders of

18-9 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions convergence for an hp-TDG√ method based on harmonic polynomials for Laplace BVPs (as opposed to exp(−b 3 #dofs) of standard schemes). The extension, using Vekua’s theory, to the Helmholtz case and plane/circular wave spaces is currently under way.

THE DISCRETE MAXIMUM PRINCIPLE IN THE FAMILY OF MIMETIC FINITE DIFFERENCE DISCRETIZATIONS Daniil Svyatskiya, Konstantin Lipnikovb and Gianmarco Manzinic

Applied Mathematics and Plasma Physics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. [email protected] [email protected] [email protected]

The Maximum Principle is one of the most important properties of solutions of elliptic partial differential equations. Its numerical analog, the Discrete Maximum Principle (DMP), is one of the properties that are very difficult to incorporate into numerical methods, especially when the computational mesh contains distorted or degenerate cells or the problem coefficients are heterogeneous and anisotropic. To mimic this property in numerical simulations is very desirable in wide range of applications. Violation of the DMP leads to non-physical solutions with numerical artifacts, such as a heat flow from a cold material to a hot one. These oscillations can be significantly amplified by non-linearity of physics. Unfortunately, numerical schemes satisfying the DMP impose severe limitations on mesh geometry and problem coefficients and often are not applicable beyond simplicial meshes. The family of the Mimetic Finite Difference (MFD) methods provides flexibility in the choice of parameters which define a particular member of the family. For example, the MFD discretization scheme for quadrilateral meshes depends on three parameters. It is pertinent to note that these parameters are chosen locally and depend on geometry and material properties in a particular cells. The correct choice of these parameters may guarantee that the resulting numerical scheme satisfy the DMP principle. The analysis of this strategy is based on the properties of M-matrices. The monotonicity limits of MFD method are investigated in several practically important cases including meshes generated using the Adaptive Mesh Refinement (AMR) strategy.

18-10 18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa- tions

A TWO-LEVEL METHOD FOR MIMETIC FINITE DIFFERENCE DISCRETIZATIONS OF ELLIPTIC PROBLEMS Marco Verani

MOX-Department of Mathematics, Politecnico di Milano, Milano, Italy [email protected]

Nowadays, the mimetic finite difference (MFD) method has become a very popular numerical approach to successfully solve a wide range of problems. This is undoubtedly connected to its great flexibility in dealing with very general polygonal meshes and its capability of preserving the fundamental properties of the underlying physical and mathematical models. In this talk, we present and analyze a two level method for mimetic finite difference discretizations of elliptic problems. The method, which is based on the introduction of suitable prolongation and restriction operators acting on polygonal meshes, is proved to be convergent. Several numerical experiments assess the efficacy of the proposed method and confirm the results of the theoretical analysis. This is a joint work with Paola F. Antonietti (Politecnico di Milano) and Ludmil Zikatanov (Penn State University).

18-11 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics 19 Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

Organisers: Georgy Kostin and Vasily Saurin

19-1 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

NORM-OPTIMAL ITERATIVE LEARNING CONTROL FOR A HEATING ROD BASED ON THE METHOD OF INTEGRO-DIFFERENTIAL RELATIONS Harald Aschemanna, Dominik Schindeleb and Andreas Rauhc

Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany. [email protected], [email protected], [email protected]

In this contribution, a multi-variable norm-optimal iterative learning control (NOILC) is designed for a spatially one-dimensional heating rod. This application represents a distributed parameter system with two Peltier elements as control inputs. For sys- tem modelling, the Method of Integro-Differential Relations (MIDR) is applied with Bernstein polynomials as ansatz functions [1]. It leads to an approximation in form of a linear state-space representation that fulfils the corresponding (energy) conser- vation law exactly. Moreover, the resulting approximation quality of the distributed parameter system can be assessed by evaluating an error measure for the constitutive relations. Considering the heat transfer in the heating rod, the constitute relation is given by Fourier’s law for heat conduction in longitudinal direction, whereas the first law of thermodynamics represents the conservation law. The repetitive control task consists in tracking trajectories – which are repeated periodically – for the desired temperature at two selected points of the heating rod by actuating two Peltier elements as control inputs. The norm-optimal iterative learning control law, cf. [2], makes use of both the tracking error of the previous iteration and temperature measurements of the current iteration to reduce the tracking error from iteration to iteration. Applying the MIDR approach, the system model results in a ninth-order state-space model that is discretized w.r.t. time using the explicit Euler method. Simulation results point out the benefits of the iterative learning control approach. The chosen NOILC design involves a combination of feedforward and feedback control and guarantees a fast convergence of the tracking errors despite model uncertainties and unknown disturbances due to convective heat losses.

References

[1] Rauh, Andreas; Senkel, Luise; Aschemann, Harald; Kostin, Georgy V.; Saurin, Vasily V.: Reliable Finite-Dimensional Control Procedures for Distributed Pa- rameter Systems with Guaranteed Approximation Quality, Proc. of IEEE Multi- Conference on Systems and Control, Dubrovnik, Croatia, 2012.

[2] Schindele, Dominik; Aschemann, Harald: Norm-Optimal Iterative Learning Control for a High-Speed Linear Axis with Pneumatic Muscles, Proc. of 8th IFAC Sympo- sium on Nonlinear Control Systems (NOLCOS), Bologna, Italy, pp. 463-468, 2010.

19-2 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

VARIATIONAL FORMULATIONS OF INVERSE DYNAMICAL PROBLEMS IN LINEAR ELASTICITY Georgy Kostina and Vasily Saurinb

Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia. [email protected], [email protected]

This contribution presents the method of integro-differential relations (MIDR) for initial-boundary value problems in the linear theory of elasticity. The main idea of MIDR is that the constitutive laws (stress-strain and momentum-velocity relations) are specified by an integral equalities instead of their local forms [1]. The modified prob- lem is reduced to minimization of a nonnegative energy error functional over admissible momentum, displacement, and stress fields under equilibrium, kinematic, initial, and boundary constraints. Based on the MIDR a parametric family of quadratic constitutive functionals are introduced and corresponding variational formulations are presented. It is shown that the conventional dual Hamilton principles result from one of the variational statements introduced. Numerical algorithms for direct and inverse dynamical problems are developed based on the Ritz method and finite element technique with spline approximations of the unknown functions in the space-time domain. The energy error functional is used to design integral criteria of the solution quality relying on the extremal proper- ties of finite-dimensional variational problems. The efficiency of the estimates proposed are demonstrated on the example of con- trolled motions for an elastic body. The problem of its displacement from the initial deformed state to the terminal one with the minimal total mechanical energy is consid- ered. The piecewise polynomial control of longitudinal displacements of a rectilinear thin rod is investigated. After FEM discretization, the original optimization problem is reduced to successive solving of two linear algebraic systems. The obtained numerical results are analyzed and discussed.

References

[1] Kostin G.V., Saurin V.V. Integrodifferential relations in linear elasticity. De Gruyter Studies in Mathematical Physics 10. De Gruyter, Berlin, 2012.

19-3 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

DESIGN AND EXPERIMENTAL VALIDATION OF CONTROL STRATEGIES FOR A SPATIALLY TWO-DIMENSIONAL HEAT TRANSFER PROCESS BASED ON THE METHOD OF INTEGRO-DIFFERENTIAL RELATIONS Andreas Rauha, Luise Senkelb and Harald Aschemannc

Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany. [email protected], [email protected], [email protected]

In previous work, different procedures have been developed for closed-loop control, ob- server design and optimization of distributed parameter systems on the basis of the Method of Integro-Differential Relations (MIDR) [1,2]. Compared to other modeling approaches for distributed parameter systems, the MIDR is characterized by its advan- tageous property of being able to quantify the approximation quality of a distributed parameter system, typically given by a set of partial differential equations, in terms of an error measure for the constitutive relations. In addition, the state variable of the distributed system is approximated in such a way that the corresponding (energy) conservation law is fulfilled exactly. In the case of heat transfer processes, the constitute relations are given by Fourier’s law in each relevant space direction. Moreover, the conservation law corresponds to the first law of thermodynamics. Here, both the heat flux density and the temperature distribution have to be approximated in a suitable form such that the above-mentioned requirements are fulfilled. In this contribution, the modeling of a spatially two-dimensional heat transfer pro- cess is described for a test rig that is available at the Chair of Mechatronics at the University of Rostock. The system consists of an aluminum plate that can be heated and cooled from one side by an array of equally large Peltier elements. This array of distributed inputs, arranged in a chess board like structure, can be used for the imple- mentation of multi-input multi-output controllers and for the realization of distributed disturbance inputs. Simulations and experimental results highlight the main properties of the MIDR which exploits Bernstein polynomials for the parameterization of both the temperature distribution and the heat flux density with an efficient formulation of inter-element conditions in a novel finite element scheme [2]. This presentation is concluded with a comparison of modeling, control and observer design that are based on either the MIDR or on a finite volume discretization of the two-dimensional heating system.

References

[1] Rauh, Andreas; Senkel, Luise; Aschemann, Harald; Kostin, Georgy V.; Saurin, Vasily V.: Reliable Finite-Dimensional Control Procedures for Distributed Pa- rameter Systems with Guaranteed Approximation Quality, Proc. of IEEE Multi- Conference on Systems and Control, Dubrovnik, Croatia, 2012.

19-4 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

[2] Rauh, Andreas; Dittrich, Christina, Aschemann, Harald: The Method of Integro- Differential Relations for Control of Spatially Two-Dimensional Heat Transfer Pro- cesses, European Control Conference ECC’13, Zurich, Switzerland, 2013. Accepted.

19-5 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

INTEGRO-DIFFERENTIAL RELATIONS IN LINEAR ELASTICITY: STATIC CASE Vasily Saurina and Georgy Kostinb

Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia. [email protected], [email protected]

Systems with distributed parameters are typically described by partial differential equa- tions (PDEs) and, in some cases, by integral or integro-differential relations. These models may also involve functionals of unknown variables. On an admissible set of functions, such a functional attains its stationary value, i.e. desired solution of the problem. This is usually associated with the problem statement based on a variational principle. One common characteristic feature inherent in variational methods is some ambigu- ity in the formulation of a finite approximation problem. It is not clear what relations are best to be weakened. As an example, the equations of linear elasticity are consid- ered. In the original statement, there are 15 variables, namely, 12 components of the stress and strain tensors as well as three components of the displacement vector, which correspond to 9 PDEs (equilibrium and kinematic equations) and 6 algebraic consti- tutive relations (Hooke’s law). If all the relations including the boundary conditions are taken in integral (weak) form then this statement coincides with the Hu-Washizu principle, which contains 18 variables (three Lagrange multipliers are added) and no constraints are imposed on them. By requiring the implementation of certain governing equations, the number of independent variables in the variational formulation can be reduced. For instance, it is possible to derive the Hellinger-Reissner principle in which there are 12 unknown functions. After successive elimination of variables, the classi- cal principle of minimum total potential energy is obtained, in which the only three variables, components of displacement vector, remain. The equivalence of these princi- ples was theoretically justified, but from a practical point of view, it is a considerable difference whether the problem is solved with respect to either 3 or 18 functions. In the numerical simulations of linear elasticity problems discussed, approximate stress and displacement fields strictly obey the equilibrium equations, kinematic rela- tions, and boundary conditions, whereas the relations of Hooke’s law are weakened, i.e., satisfied in some integral sense [1]. It looks rather reasonable in numerical realiza- tion to present Hooke’s law as an integral over a function which is a quadratic form of the stress-strain relations. The functional of energy error gives one the possibility to divide the problem originally formulated in terms of stresses and displacements into two independent subproblems: one in the displacements, the other in the stresses. For various variational formulations following the method of integro-differential relations, the bilateral energy estimates of approximate solution quality are presented. Finite element algorithms were developed not only to check for model errors but also to refine adaptively FEM meshes in order to improve the solution quality.

19-6 19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of Mathematical Physics

References

[1] Kostin G.V., Saurin V.V. Integrodifferential relations in linear elasticity. De Gruyter Studies in Mathematical Physics 10. De Gruyter, Berlin, 2012.

19-7 20: Mini-Symposium: Large scale computing with applications

20 Mini-Symposium: Large scale computing with applications

Organiser: Ulrich R¨ude

20-1 20: Mini-Symposium: Large scale computing with applications

ADAPTIVE ASYNCHRONOUS PARALLEL CALCULATIONS AT PETASCALE USING UINTAH Martin Berzins

Scientific Computing and Imaging Institute 72 S Central Campus Drive, Room 3750, Salt Lake City, UT 84112 USA [email protected]

The past, present and future scalability of the Uintah Software framework is considered with the intention of describing a successful approach to large scale parallelism. Uintah allows the solution of large scale fluid-structure interaction problems through the use of fluid flow solvers coupled with particle-finite element based solids methods. In addition Uintah uses a combustion solver to tackle a broad and challenging class of turbulent combustion problems. A unique feature of Uintah is that it uses an asynchronous task-based approach with automatic load balancing to solve complex problems using techniques such as adaptive mesh refinement. At present, Uintah is able to make full use of present-day massively parallel machines as the result of three phases of development over the past dozen years. These development phases have led to an adaptive scalable run-time system that is capable of independently scheduling tasks to multiple CPUs cores and GPUs on a node. In the case of solving incompressible low- mach number applications it is also necessary to use linear solvers and to consider the challenges of radiation problems. The approaches adopted to achieve present scalability are described and their extensions to possible future architectures is considered.

20-2 20: Mini-Symposium: Large scale computing with applications

ACCELERATOR-FRIENDLY PARALLEL ADAPTIVE MESH REFINEMENT Carsten Burstedde1, Lucas C. Wilcox2, Georg Stadler3 and Donna Calhoun4

1Institut f¨urNumerische Simulation, Universit¨atBonn, Germany [email protected] 2Department of Applied Mathematics, Naval Postgraduate School, USA 3Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA 4Boise State University, USA

Recently, we have witnessed an impressive increase in total available compute power, which is commonly measured in flops per second. We have also seen applications that sustain a rather high percentage of the theoretical peak performance. With regards to the numerical solution of partial differential equations, achieving high peak percentage is a constant struggle for multiple reasons.

1. Sparse linear algebra is hard to vectorize.

2. The strong scaling behavior of multilevel solvers is suboptimal.

3. Finite element mesh traversal causes indirections which can be slow.

For the ostensibly simple case of explicit solvers, the first item still applies regarding matrix-vector products, and the second still applies when local or hierarchical time stepping schemes are used. The third applies whenever the mesh is non-uniform or non-structured. With unstructured or tree-structured adaptive mesh refinement, the atomic mesh primitive is usually identified with a classical finite element, which entails all of the above predicaments. One way out lies in the reinterpretation of the atomic mesh primitive as a macro-element, thus introducing a two-level discretization hierarchy which is a better basis for writing optimized code. In fact, one of the earliest examples of such an approach are unstructured conforming spectral elements demonstrated by Tufo and Fischer in 1999. We have developed h-adaptive non-conforming spectral elements for the numerical solution of the elastic-acoustic wave equation in the past, which were subsequently ported to multi-GPU clusters, applying MPI parallelization for the adaptive mesh and thread-level parallelization across the individual degrees of freedom of each high-order spectral element. Currently, we are investigating a new approach to solving transport problems, namely using uniform finite volume subgrids as the leafs of a tree-based adaptively refined mesh. Work is underway to use thread-level parallelization on the subgrids as well. We will explain the details of our approach to cache- and accelerator-friendly scal- able adaptive mesh refinement and present numerical results for the solution of both hyperbolic and parabolic partial differential equations.

20-3 20: Mini-Symposium: Large scale computing with applications

TOTAL EFFICIENCY OF CORE COMPONENTS IN FINITE ELEMENT FRAMEWORKS Markus Geveler

Institute for Applied Mathematics and Numerics, TU Dortmund, Germany [email protected]

Various techniques to exploit heterogeneous computational hardware in clusters are used for large scale Continuum Mechanics simulations. We restrict ourselves to Fluid Dynamics / Structure Mechanics solvers and concentrate on core components of Fi- nite Element frameworks like, e.g. the linear solvers therein. The target hardware ranges from clusters comprising Multicore-CPU / (multi-)GPU nodes to an experi- mental ARM-based cluster. Hardware-efficiency aspects (exploitation of all levels of parallelism and accelerator hardware) as well as the numercial efficiency (using ad- justed Finite Element geometric Multigrid solvers over minor solvers) are considered and finally, energy-efficiency is discussed briefly.

MASSIVE PARALLEL SIMULATION OF WATER AND SOLUTE TRANSPORT IN POROUS MEDIA Olaf Ippischa, Markus Blatt and Jorrit Fahlke

Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany [email protected]

Water flow and solute transport are topics of high relevance for many problems with high importance for society. The multi-scale heterogeneity of natural porous requires large scale simulations with high spatial resolution. Water flow in partially saturated porous media is described by Richards’ equation, a non-linear degenerate parabolic partial differential equation of second-order. The numerical solution is based on a cell-centred Finite-Volume discretisation, an implicit Euler-scheme in time, linearisation with an inexact Newton scheme with line search and solution of the linear equation system with a BiCGStab solver preconditioned by algebraic multi-grid. A convection-diffusion equation is used for solute transport. As the flow field can be quite heterogeneous and convection dominant, an explicit second order Godunov scheme with a minmod slope limiter is used for the convective part and a Finite-Volume discretisation for the diffusive part. The results of scalability tests with up to 150 billion unknowns on 294849 cores of the Bluegene/P type super computer JUGENE for the linear solver and the complete model including I/O are presented as well as results from practical applications.

20-4 20: Mini-Symposium: Large scale computing with applications

SCALABLE PARALLEL MULTILEVEL SOLUTION OF ELLIPTIC PROBLEMS Peter K. Jimack1a, Mark A. Walkley1b and Jianfei Zhang2

1School of Computing, University of Leeds, Leeds LS2 9JT, UK [email protected], [email protected] 2Department of Engineering Mechanics, Hohai University, 1 Xikang Road, Nanjing, China 210098 [email protected]

Multilevel solvers, such as geometric or algebraic multigrid, have been applied to the optimal solution of algebraic systems arising from the discretization of elliptic partial differential equations (PDEs) for a number of decades. For standard finite difference and finite element discretizations this optimality allows an accurate solution to be obtained at a cost of O(N) operations, where N is the number of degrees of freedom. The challenges of implementing such solvers on large scale parallel systems stem from the fact that the computations that must be undertaken at the coarsest levels do not generally contain sufficient work to hide or offset the parallelization overhead. In this work we consider the parallel solution of a scalar PDE and of systems of linear elasticity equations using a parallel algebraic multigrid preconditioner [A. Napov and Y. Notay, SIAM J. Sci. Comput., 34, 1079-1109, 2012]. In particular, we investigate the effects of applying different approximations at the coarsest levels in order to obtain the best parallel performance on large-scale systems. Numerical examples will be presented to show that the best results are obtained when the accuracy of the coarse grid solve is sacrificed for improved parallel performance.

20-5 20: Mini-Symposium: Large scale computing with applications

THOUGHTS ON GENERAL PURPOSE FINITE ELEMENT LIBRARIES AND HYBRID PROGRAMMING Guido Kanschat

Interdisziplin¨aresZentrum f¨urWissenschaftliches Rechnen (IWR), Universit¨atHeidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg Germany [email protected]

Modern high performance architectures are characterized through a hierarchy of paral- lel computing units, from multicore processors, which share memory in a single physical computing node equipped with GPU boards to clusters of thousands of such nodes and heterogeneous networks. The challenge originating from these hardware structures for developers of a general purpose finite element library consists in the fact that the gap between achieved and peak performance widens more and more if software structures do not take these into account. Therefore, a balance has to be found between wide applicability of the library and sufficient hardware adaptation. Accordingly, certain structures and algorithms we have grown fond of will have to be replaced by more effi- cient or robust ones. We discuss the high performance computing support existing in the deal.II library, its limits, and future developments to overcome current deficiencies.

PATCHING ADAPTIVITY FOR LARGE SCALE PROBLEMS - A NEW LIGHTWEIGHT ADAPTIVE SCHEME AND ITS APPLICATION IN COMPUTATIONAL ELECTROCARDIOLOGY Dorian Krause, Rolf Krausea, Thomas Dickopf and Mark Potse

Institute of Computational Science, Universit`adella Svizzera italiana, Lugano, Switzerland [email protected]

Adaptive strategies are well known to reduce the number of degrees of freedom sub- stantially. However, in case of parallel large scale computations even a significant reduction in degrees of freedom does not necessarily lead to a similarly significant re- duction of overall computing time. This is mostly due to the associated overhead, which originates from error estimation, mesh adaptation, mesh redistribution, load balancing, and the use of the associated complicated data structures. Here, we present a novel lightweight adaptive approach, which combines locally structured meshes with a non-conforming mortar discretisation. Our approach aims at combining the power of adaptivity with computational efficiency, parallel scalability, and implementational sim- plicity. We present and analyse this new method in the framework of the monodomain equation from computational electrocardiology along with problems of different sizes. As it turns out, our lightweight adaptive scheme provides a very good balance between reduction in degrees of freedom and overall (parallel) efficiency.

20-6 20: Mini-Symposium: Large scale computing with applications

FAST AND SCALABLE ELLIPTIC SOLVERS FOR ANISOTROPIC PROBLEMS IN GEOPHYSICAL MODELLING Eike Mueller1a, Robert Scheichl1b and Eero Vainikko2

1 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. [email protected], [email protected]. 2 Faculty of Mathematics and Computer Science, University of Tartu, Liivi 2, Tartu 50409, Estonia. [email protected]

Semi-implicit time stepping is very popular and widely used in numerical weather- and climate prediction models for various reasons, particularly since it allows for larger time steps and thus for better efficiency. However, the bottleneck in semi-implicit schemes is the need for a three dimensional elliptic solve in each time step. With increasing model resolution this elliptic PDE can only be solved on operational timescales if highly efficient algorithms are used and their scalability and performance on modern computer architectures can be guaranteed. We have studied a typical model equation arising from semi-implicit semi-Lagrangian time stepping; problems with a similar structure are encountered in other areas of geo- physical modelling, such as ocean circulation models and subsurface flow simulations. In particular, the vertical extent of the domain is significantly smaller than the hor- izontal size and one of the definining characteristics of elliptic PDEs encountered in geophysical applications is a strong anisotropy in the vertical direction. To take this into account, these equations are usually discretised on grids with a tensor-product structure with a unstructured (or semi-structured) horizontal mesh and a regular grid for each vertical column. We implemented a bespoke, matrix-free geometric multigrid solver based on [B¨ormS., Hiptmair R., Numer. Algorithms 26: 200-1. (1999)] and [Buckeridge S, Scheichl R., Numerical Linear Algebra with Applications 17(2-3): 325– 342 (2010)] which exploits the grid structure and strong vertical anisotropy and avoids precomputation of the matrix and coarse grid setup costs. We demonstrated the superior performance of our solver on grids with a tensor product structure by comparing it to existing AMG solvers from the DUNE and Hypre libraries and showed its scalability and robustness for systems with more than 1010 degrees of freedom on up to 65536 CPU cores of the HECToR supercomputer. To investigate the performance on non-standard chip architectures, we ported the solver to a cluster of Graphics Processing Units (GPUs) using the CUDA-C programming model. As the calculation is memory bound, the implementation was optimised to account for the GPU specific memory hierachy and the limited bandwidth of host- device data transfers.

20-7 20: Mini-Symposium: Large scale computing with applications

PARALLEL INCOMPRESSIBLE FLOW SIMULATIONS USING DIVERGENCE-FREE FINITE ELEMENTS Tobias Neckel

Department of Informatics, Technische Universit¨atM¨unchen Boltzmannstraße 3, 85748 Garching, Germany [email protected]

Finite Elements have proven to be a valuable approach for numerical discretisation in various fields of applications. In the case of incompressible flow simulations, different variants exist of how to combine a spatial discretisation (via FEM, e.g.) of the under- lying Navier-Stokes equations with integration in time: explicit methods, fully implicit discretisations, stabilised formulations, etc. The crucial point in all approaches is how to respect the continuity equation. We developed a specific type of divergence-free elements in 2D (see [1, 4, 5]) which exactly (i.e. continuously) fulfil the continuity equation. This enables a simultaneous conservation of momentum and energy, a feature that established element types typ- ically do not possess (cf. [3]) but that is advantageous for various applications such as turbulent flow scenarios or coupled simulations. One important advantage of such divergence-free elements is the resulting skew-symmetric convection operator in the dis- cretisation. The divergence-free elements are similar to the well-known Q1Q0 elements both with respect to their low order and the corresponding computational costs. In this contribution, we present the extension of the divergence-free finite elements to the three-dimensional case. We use this spatial discretisation in a semi-implicit Chorin projection scheme. The implementation has been realised in the PDE frame- work Peano [2] which uses (adaptive) Cartesian grids in combination with space-filling curves and stack data structures for efficient simulations. Due to the low order of the elements and the form of the grid cells, we precompute element matrices for all op- erators instead of performing numerical quadrature. Different benchmark simulations are used to verify the code and show the validity of the approach. Furthermore, we discuss certain aspects of the parallelisation of our discretisation. Applying a domain decomposition for Cartesian grids, we simulated porous media-like scenarios (fracture networks in 2D and sphere packings in 3D) with several hundreds of CPUs on different computing environments. The focus of current work is on the implementation and verification of parallel flow simulations using fully adaptive grids as well as on applying high numbers of processors.

References

[1] C. Blanke. Kontinuit¨atserhaltende Finite-Element-Diskretisierung der Navier- Stokes-Gleichungen. Diploma Thesis, Technische Universit¨atM¨unchen, 2004.

[2] H.-J. Bungartz, M. Mehl, T. Neckel, and T. Weinzierl. The PDE framework Peano applied to fluid dynamics: An efficient implementation of a parallel multiscale fluid

20-8 20: Mini-Symposium: Large scale computing with applications

dynamics solver on octree-like adaptive Cartesian grids. Computational Mechanics, 46(1):103–114, June 2010.

[3] P. M. Gresho and R. L. Sani. Incompressible Flow and the Finite Element Method. John Wiley & Sons, 1998.

[4] T. Neckel. The PDE Framework Peano: An Environment for Efficient Flow Sim- ulations. Verlag Dr. Hut, 2009.

[5] T. Neckel, M. Mehl, and C. Zenger. Enhanced divergence-free elements for efficient incompressible flow simulations in the PDE framework Peano. In Proceedings of the Fifth European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2010, 14th-17th June 2010, Lissabon, 2010.

ON LARGE-SCALE MECHANICS SIMULATIONS WITH THE PARALLEL TOOLBOX Aurel Neic

University of Graz, Institute of Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria [email protected]

The Cardiac Arrhythmias Research Package [G. Plank] simulates the electric and me- chanic behaviour of the heart tissue. Since the goal of this package is to resolve the anatomic properties of the heart in high detail, the FE discretisations usually consist of several millions of elements. This is a computationally challenging problem. While originally relying on PETSc for the data framework and solvers, CARP recently incor- porated the Parallel Toolbox for its GPU accelerated AMG-PCG solver. In this talk I will highlight the algorithmic and computational challenges in large-scale linear and non-linear mechanics simulations on clusters of GPUs and CPUs. Additionally I will present new developments in the domain-decomposition parallelization techniques, such as minimizing computational imbalances related to unbalanced subdomain boundaries and reducing the communication complexity of the Parallel Toolbox.

20-9 20: Mini-Symposium: Large scale computing with applications

RECENT DEVELOPMENTS IN NGSOLVE FOR DISTRIBUTED AND MANY-CORE PARALLEL COMPUTING Joachim Sch¨oberl

Institute for Analysis and Scientific Computing, Vienna UT, Austria [email protected]

NGSolve is a general purpose high-order finite element package. Available application classes are, among others, electromagnetics, wave propagation, and incompressible flows. Numerical techniques include H(curl) and H(div) elements, hybrid-DG methods, and according preconditioning techniques. One part of the talk reports on the distributed memory parallelization. We dis- cuss the rather small programming interface of the parallel layer to the numerical- components layer. We report on the performance of high-order domain decomposition methods, in combination with direct and amg-based coarse grid preconditioners. In the second part we discuss current software design for many core (GPGPU) hardware. Current hardware architecture providing coarse-grain and fine-grain paral- lelism is very attractive for high order finite element methods, which are more compute intensive than memory bandwidth consuming.

ANALYSIS OF ADAPTIVE SPACE-TIME FINITE ELEMENTS FOR PARABOLIC PROBLEMS Kunibert G. Siebert

Institute for Applied Analysis and Numerical Simulation, Universit¨atStuttgart, Germany [email protected]

We present an adaptive space-time finite element method for parabolic problems. The algorithm is based on a classical adaptive time-stepping scheme supplemented by an additional control of a potential energy increase of the discrete solution originating from coarsening of the spatial meshes. We show that this algorithm is converging. This means, given a positive tolerance, the algorithm reaches final time in a finite number of steps and with an adaptive choice of the spatial meshes and the time-step-sizes such that the total space-time error is below the given tolerance. This is joint research with Christian Kreuzer (Bochum), Christian A. M¨oller(Augs- burg), and Alfred Schmidt (Bremen).

20-10 20: Mini-Symposium: Large scale computing with applications

SCALABLE SOLVERS FOR ELLIPTIC PROBLEMS DISCRETIZED BY ADAPTIVE HIGH-ORDER FINITE ELEMENTS Georg Stadler1a, Tobin Isaac1b, Hari Sundar1c, Carsten Burstedde2 and Omar Ghattas1e

1Institute for Computational Engineering and Sciences The University of Texas at Austin, Austin, TX USA [email protected], [email protected], [email protected] [email protected] 2Institute for Numerical Simulation, University of Bonn, Germany [email protected]

I will discuss challenges that arise in the solution of large-scale elliptic partial differ- ential equations discretized by higher-order adaptive finite elements. As examples, an equation with a positive definite operator as well as the Stokes equation will be con- sidered. The efficiency of iterative Krylov methods for the solution of these systems relies on the availability of scalable preconditioners, for which we consider algebraic and geometric multigrid methods. We compare the scalability of these methods to tens of thousands of CPU cores and discuss their ability to handle higher-order discretizations.

ALGEBRAIC MULTILEVEL PRECONDITIONING IN H(CURL) AND H(DIV) SPACE Satyendra Tomar

RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria [email protected]

An algebraic multilevel iteration method for solving system of linear algebraic equa- tions arising in H(curl) and H(div) space will be presented. The algorithm is developed for the discrete problem obtained by using the space of lowest order Raviart-Thomas- Nedelec elements of H(curl) and H(div). The theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local (element- wise) problems. Explicit recursion formulae are derived to compute the element ma- trices and the constant γ (which measures the quality of the space splitting) at any given level. It will be proved that the proposed method is robust with respect to the problem parameters, and is of optimal order complexity. Supporting numerical results, including the case when the parameters have jumps, will also be presented.

20-11 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

21 Mini-Symposium: Low Rank Tensor Based Nu- merical Methods

Organisers: Lars Grasedyck, Boris Khorom- skij and Dmitry Savostyanov

21-1 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

ADAPTIVE METHODS BASED ON TENSOR REPRESENTATIONS OF COEFFICIENT SEQUENCES AND THEIR COMPLEXITY ANALYSIS Markus Bachmayr and Wolfgang Dahmen

Institut f¨urGeometrie und Praktische Mathematik, RWTH Aachen, Germany. [email protected]

We consider a framework for the construction of iterative schemes for high-dimensional operator equations that combine adaptive approximation in a basis and low-rank ap- proximation in tensor formats. Our starting point is an operator equation Au = f, where A is a bounded and elliptic linear operator mapping a separable Hilbert space H – for instance, a function space on a high-dimensional product domain – to its dual H0. Assuming that a Riesz basis of H is available, the original problem can be rewritten as a linear system on `2, where the system matrix is bounded and continously invertible. Under the given assumptions, a simple Richardson iteration on the infinite-dimensional problem converges, but of course cannot be realized in practice. This is the starting point for adaptive wavelet methods as introduced by Cohen, Dahmen and DeVore, which dynamically approximate such an ideal iteration by finite quantities, exploiting the approximate sparsity of coefficient sequences. The new aspect here is that, in order to significantly reduce computational com- plexity in a high dimensional context, we make use of an additional tensor product structure of the problem. For this discussion, we assume H = H1 ⊗ · · · ⊗ Hd, i.e., that H is a tensor product Hilbert space, and that we have a tensor product Riesz basis of H. We now use a structured tensor format for the corresponding sequence of basis coefficients. Examples of suitable tensor structures are the Tucker format or the Hierarchical Tucker format, where the latter can also be used for problems in very high dimensions. A crucial common feature of both formats is that there exist reliable pro- cedures for obtaining quasi-best approximations by lower-rank tensors with controlled error in `2-norm. We are thus considering a highly nonlinear type of approximation: besides the multiplicative nonlinearity in the tensor representation, we aim to adaptively determine simultaneously suitable finite approximation ranks, the active indices for the basis expansions in the lower-dimensional spaces Hi, and corresponding coefficients. We accomplish this by a perturbed Richardson iteration, where approximation ranks and active basis indices are adjusted implicitly in a sufficiently accurate approximation of the residual. The resulting growth in the complexity of iterates is kept in check by combining a tensor recompression operation, which yields an approximation with lower ranks up to a specified error, with a coarsening operation that eliminates negligible coefficients in the lower-dimensional basis expansions. In the efficient realization of the latter, the special orthogonality properties of the considered tensor formats play a central role. Under the present quite general assumptions, we can then identify a choice of pa- rameters for the resulting iterative scheme that ensures its convergence and produces approximations with near-minimal ranks. Under suitable further approximability con-

21-2 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods ditions on the problem, we also obtain estimates for the total number of operations required for reaching an approximate solution with a certain target accuracy. Further- more, we discuss the additional difficulties related to the preconditioning of problems posed on Sobolev spaces in this setting. We consider some possible applications and illustrate our theory by numerical experiments.

BLACK BOX APPROXIMATION STRATEGIES IN THE HIERARCHICAL TENSOR FORMAT Jonas Ballania and Lars Grasedyckb

Institut f¨urGeometrie und Praktische Mathematik, RWTH Aachen, Germany. [email protected], [email protected]

The hierarchical tensor format allows for the low-parametric representation of tensors even in high dimensions d. The efficiency of this representation strongly relies on an appropriate hierarchical splitting of the different directions 1, . . . , d such that the associated ranks remain sufficiently small. This splitting can be represented by a binary tree which is usually assumed to be given. In this talk, we address the question of finding an appropriate tree from a subset of tensor entries without any a priori knowledge on the tree structure. The proposed strategy can be combined with rank- adaptive cross approximation techniques such that tensors can be approximated in the hierarchical format in an entirely black box way. Numerical examples illustrate the potential and the limitations of our approach.

21-3 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS. PART II: FASTER ALGORITHM AND APPLICATION TO NONSYMMETRIC SYSTEMS Sergey V. Dolgov1 and Dmitry V. Savostyanov2

1Max Planck Institute for Mathematics in Sciences, Leipzig, Germany, [email protected] 2School of Chemistry, University of Southampton, UK, [email protected]

In this talk we further develop and investigate the rank-adaptive alternating methods for high-dimensional tensor-structured linear systems. The ALS method is reformu- lated in a recurrent variant, which performs a subsequent linear system reduction, and the basis enrichment is derived in terms of the reduced system. This algorithm appears to be more robust than the method based on a global steepest descent correction, and additional heuristics allow to speedup the computations. Furthermore, the very same method is applied to nonsymmetric systems as well. Though its theoretical justification is based on the FOM method, and is more difficult than in the SPD case, the practical performance is still very satisfactory, which is demonstrated on several examples of the Fokker-Planck and chemical master equations. Keywords: high–dimensional problems, tensor train format, ALS, DMRG, steepest descent, convergence rate, superfast algorithms, NMR. References: arXiv:1301.6068[math.NA], arXiv:1304.1222[math.NA]

21-4 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

HP -DG TIME STEPPING FOR HIGH-DIMENSIONAL EVOLUTION PROBLEMS WITH LOW-RANK TENSOR STRUCTURE Vladimir Kazeev

Seminar for Applied Mathematics, ETH Z¨urich, CH-8092 Z¨urich, Switzerland [email protected]

We consider linear evolution equations posed in spaces of possibly high dimensions. Im- portant examples are Kolmogorov equations: Fokker–Planck equations for stochastic ODEs in finance and the Chemical Master Equation in chemistry and systems biol- ogy. While the latter often exhibit dynamics with pronounced transient phases, the former may essentially require adaptivity to allow for time-inhomogeneous degenerate diffusion. Moreover, both tend to have high spatial dimension, which makes the use of classical adaptive or explicit full tensor-product discretizations in space unfeasible. For equations combining such features we propose an approach based on the hp-DG discretization in time and the low-rank tensor approximation in space. In a suitable time-weighted Bochner space, a time-inhomogeneous degenerate evo- lution problem is shown to be well-posed and the analytic regularity of the time- dependence of the solution is studied. The hp-discontinuous Galerkin time discretiza- tion is shown to converge exponentially with respect to the number of temporal degrees of freedom. To overcome the “curse of dimensionality” in space, the Tensor Train (TT) repre- sentation of high-dimensional arrays, based on the separation of variables, is employed. The low-rank approximation of a tensor in the TT format can be performed with the use of standard, well-established matrix algorithms. On the other hand, for many ap- plications, the complexity and memory requirements are linear or almost linear with respect to the number of dimensions. The use of quantization, leading to the QTT de- composition, allows to resolve even more structure in the matrices and vectors involved and to further reduce the complexity and memory requirements. Numerical experiments with equations of the two types mentioned above demon- strate the efficientcy of the proposed approach.

21-5 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

HARTREE-FOCK EIGENVALUE SOLVER USING TENSOR-STRUCTURED TWO-ELECTRON INTEGRALS Venera Khoromskaia

Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany [email protected]

The Hartree-Fock eigenvalue problem with the 3D integro-differential operator repre- sents the basic model in ab-initio electronic structure calculations. Due to presence of multiple strong cusps in electron density of molecules, the traditional approach to the solution of the Hartree-Fock equation is based on accurate analytical precomputation of the arising 6D convolution type integrals with the Newton kernel, the so-called two- electron integrals (TEI), using the naturally separable Gaussian-type basis functions. We present an alternative, fast “black-box“ Hartee-Fock solver by the tensor numer- ical methods based on the rank-structured calculation of the core Hamiltonian and TEI using a general, well separable basis discretized on n × n × n 3D Cartesian grid [3, 4, 5, 7]. The arising 6D convolution integrals are approximated by 1D algebraic operations in O(n log n) complexity on large spatial grids up to the size n3 ≈ 1014, thus providing high resolution of molecular cusps at low cost [1, 2]. The truncated Cholesky decomposition of TEI matrix is based on multiple factorizations, including the purely algebraic directional “1D density fitting“ depending on a threshold ε > 0, yielding an almost irreducible number of product basis functions for building the TEI tensor [5]. The factorized TEI matrix is applied in tensor-based post-Hartree-Fock cal- culations [6]. We present on-line (in Matlab) ab initio ground state energy calculations for compact molecules, including glycine and alanine amino acids [7].

References

[1] B. N. Khoromskij and V. Khoromskaia. Multigrid Tensor Approximation of Func- tion Related Arrays. SIAM J Sci. Comp., 31(4), 3002-3026, 2009.

[2] V. Khoromskaia. Computation of the Hartree-Fock Exchange in the Tensor- structured Format. CMAM, Vol. 10, No 2, pp.204-218, 2010.

[3] B. N. Khoromskij, V. Khoromskaia and H.-J. Flad. Numerical solution of the Hartree-Fock equation in Multilevel Tensor-structured Format. SIAM J Sci. Comp., 33(1), 45-65, 2011.

[4] V. Khoromskaia, D Andrae and B.N. Khoromskij. Fast and Accurate Tensor Cal- culation of the Fock Operator in a General Basis. CPC, 183, 2392-2404, 2012.

[5] V. Khoromskaia, B.N. Khoromskij and R. Schneider. Tensor-structured Calcula- tion of the Two-electron Integrals in a General Basis. Preprint 29/2012 MIS MPI, Leipzig, 2012. SIAM J. Sci. Comp., 2013, to appear.

21-6 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

[6] V. Khoromskaia, and B.N. Khoromskij. Møller-Plesset Energy Correction Using Tensor Factorizations of the Grid-based Two-electron Integrals. Preprint 26/2013 MIS MPI Leipzig, 2013.

[7] V. Khoromskaia. Fast 3D grid-based Hartree-Fock solver by tensor methods. In progress, 2013.

SUPER-FAST SOLVERS FOR PDES DISCRETIZED IN THE QUANTIZED TENSOR SPACES Boris Khoromskij

Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany [email protected]

Tensor numerical approximation provides the efficient separable representation of mul- tivariate functions and operators on large n⊗d-grids, that allows the solution of d- dimensional PDEs with linear complexity scaling in the dimension, O(dn). Modern methods of separable approximation combine the canonical, Tucker, as well as the ma- trix product state (MPS) and tensor train (TT) formats [2]. The recent quantized TT (QTT) approximation [1] is proven to provide the logarithmic data-compression on a wide class of functions and operators. It makes possible to solve high-dimensional steady-state and dynamical problems in quantized tensor spaces, with the log-volume complexity scaling in the full-grid size, O(d log n), instead of O(nd). In this talk I show how the grid-based QTT tensor approximation applies to hard problems arising in electronic structure calculations, such as many-electron integrals [3]. The QTT approximation method provides the efficient solvers for parametric PDEs [4] as well as for high-dimensional time-dependent models, in particular, the molecular Schr¨odinger,the Fokker-Planck [5] and chemical master equations [6]. We present numerical tests indicating the efficiency of the QTT tensor approximation in some steady-state and dynamical calculations.

References

[1] B.N. Khoromskij. O(d log N)-Quantics Approximation of N-d Tensors in High- Dimensional Numerical Modeling. J. Constr. Approx. v. 34(2), 257-289 (2011).

[2] B.N. Khoromskij. Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances. Chemometr. Intell. Lab. Syst. 110 (2012), 1-19. DOI: 10.1016/j.chemolab.2011.09.001.

[3] V. Khoromskaia, B.N. Khoromskij, and R. Schneider. Tensor-structured calcula- tion of two-electron integrals in a general basis. SIAM J. Sci. Comput., 2013 (to appear). Preprint 29/2012, MPI MiS, Leipzig 2012.

[4] B.N. Khoromskij, and Ch. Schwab, Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs. SIAM J. Sci. Comp., 33(1), 2011, 1-25.

21-7 21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

[5] S.V. Dolgov, B.N. Khoromskij, and I. Oseledets. Fast solution of multi-dimensional parabolic problems in the TT/QTT formats with initial application to the Fokker- Planck equation. SIAM J. Sci. Comput., 34(6), 2012, A3016-A3038.

[6] S. Dolgov, and B.N. Khoromskij. Tensor-product approach to global time-space- parametric discretization of chemical master equation. Preprint 68/2012, MPI MiS, Leipzig 2012 (submitted).

ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS. PART I: SPD SYSTEMS Dmitry V. Savostyanov1 and Sergey V. Dolgov2

1School of Chemistry, University of Southampton, UK, [email protected] 2Max Planck Institute for Mathematics in Sciences, Leipzig, Germany, [email protected]

We propose a new algorithm for the approximate solution of large–scale high–dimensio- nal tensor–structured linear systems. It can be applied to high-dimensional differential equations, which allow a low–parametric approximation of the multilevel matrix, right hand side and solution in the tensor train format. We combine the Alternating Linear Scheme approach with the basis enrichment idea using Krylov–type vectors. We obtain the rank–adaptive algorithm with the theoretical convergence estimate not worse than the one of the steepest descent. The practically observed convergence is significantly faster, comparable or even better than the convergence of the DMRG–type algorithm. The complexity of the method is at the ALS level. The method is successfully applied for a high–dimensional problem of quantum chemistry, namely the NMR simulation of a large peptide. Keywords: high–dimensional problems, tensor train format, ALS, DMRG, steepest descent, convergence rate, superfast algorithms, NMR. References: arXiv:1301.6068[math.NA], arXiv:1304.1222[math.NA]

21-8 22: Mini-Symposium: Mathematical and statistical modeling in biology

22 Mini-Symposium: Mathematical and statistical modeling in biology

Organiser: H.T. Banks

22-1 22: Mini-Symposium: Mathematical and statistical modeling in biology

ACOUSTIC LOCALISATION OF CORONARY ARTERY STENOSIS: WAVE PROPAGATION IN SOFT TISSUE MIMICKING GEL H. Thomas Banks4, Malcolm J. Birch2, Mark P. Brewin1,2, Steve E. Greenwald1a, Shuhua Hu4, Zackary Kenz4, Carola Kruse3, Dwij Mehta1b, Simon Shaw3 and John R. Whiteman3

1Blizard Institute, Barts and The London School of Medicine and Dentistry, Queen Mary, University of London, UK [email protected], [email protected] 2Clinical Physics, Barts Health NHS Trust, London, UK 3BICOM, Institute of Computational Mathematics, Brunel University, UK 4CRSC, Department of Mathematics, North Carolina State University, Raleigh NC, USA

Plaque developing in a coronary artery produces turbulent flow downstream and wall shear stresses varying at a frequency around 1kHz. These give rise to low amplitude acoustic shear waves which propagate through the chest and can be measured by skin sensors. This acoustic surface signature may provide a cheap non-invasive means of diagnosing arterial disease. We will discuss measurements of the propagation of 1-D free oscillations induced in tissue mimicking gel specimens following the sudden release of shear or compressive stresses and the results of subsequent measurements of the surface strain field, using a novel optical technique, due to 2-D forced oscillations, induced in the gel by an electro mechanical vibrator. We will also describe measurements of the strain field on the surface of cuboidal and cylindrical gel specimens containing unobstructed and stenosed (partially obstructed) tubes through which laminar and turbulent flow is passed. A companion presentation will describe the results of direct and inverse solver software to simulate the response of the gel to the shear waves. This mathematical loop makes it possible to characterise the source given the signal and to compare material data with predicted values.

22-2 22: Mini-Symposium: Mathematical and statistical modeling in biology

EFFICIENT NUMERICAL METHODS FOR COUPLED PDE-ODE SYSTEMS: AN APPLICATION IN INTERCELLULAR SIGNALING Thomas Carraroa, Elfriede Friedmannb and Daniel Gerechtc

Institute for Applied Mathematics, Heidelberg University, Heidelberg, Germany [email protected], [email protected] [email protected]

A combined experimental and theoretical study [Busse et al, PNAS 2010] has shown that intercellular diffusion of the cytokine Interleukin-2 (IL-2) is a key regulatory step in the induction of immune responses in the lymph node. Showing the importance of spatial distribution, it implies that a realistic 3D configuration of the cells is necessary for understanding the observed processes. We developed efficient numerical methods to solve the nonlinear system of equations consisting of a PDE coupled with ODEs. Our principal components are a Galerkin space discretization by finite elements, a fully coupled multilevel algorithm as solver and an adaptive time scheme. 3D simulations show how the competition of T helper cells and regulatory T cells for IL-2 influence the activation of the T cells. We compare results of a homogenized model to the full system.

MODELING AND INVERSE PROBLEM CONSIDERATIONS FOR A VISCOELASTIC TISSUE MODEL Zackary Kenz

Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205, U.S.A. [email protected],

Existing methods used to detect an arterial stenosis are laborious, expensive, and often invasive. A method has been proposed whereby one places sensors on the surface of the chest and attempts to detect shear waves generated by turbulent flow (generated by a blockage) impacting the vessel wall. We focus here on the propagation of shear and pressure waves through a viscoelastic medium. We will discuss the development of one-dimensional models for wave propagation in a tissue-mimicking agar gel cylinder. Corresponding experimental data from the gel has been obtained by our collaborators. We present a sample of our inverse problem model parameter results using this data, and also examine the robustness of the estimated parameters. This is work performed in collaboration with H.T. Banks and Shuhua Hu, NCSU; with Carola Kruse, Simon Shaw, and John R. Whiteman at BICOM, Brunel University, London; with Stephen E. Greenwald and Mark P. Brewin at Blizard Institute, Barts and the London School of Medicine and Dentistry, Queen Mary, University of London; and with Malcom J. Birch at Clinical Physics, Barts Health NHS Trust, London.

22-3 22: Mini-Symposium: Mathematical and statistical modeling in biology

HIGH ORDER SPACE-TIME FINITE ELEMENT SCHEMES FOR THE DYNAMICS OF VISCOELASTIC SOFT TISSUE Carola Kruse1a, Simon Shaw1, John R. Whiteman1, H. Thomas Banks2, Zackary Kenz2, Shuhua Hu2, Steve E. Greenwald3, Mark P. Brewin3 and Malcolm J. Birch4

1BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge, UK [email protected] 2Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA 3Blizard Institute, Barts and The London School of Medicine and Dentistry, Queen Mary, University of London, UK 4Clinical Physics, Barts and the London National Health Service Trust, UK

As plaque builds up in a coronary artery, blood flow past the stenosed region becomes turbulent and creates abnormal variations in wall shear stresses in the wake. These shears drive low amplitude acoustic shear waves at around 1 kHz through the soft tis- sue in the thorax which appear at the chest wall and can be measured non-invasively by placing sensors on the skin. This acoustic surface signature (bruit) has thus the po- tential to provide a cheap non-invasive means of diagnosing Coronary Artery Disease [Banks and Pinter, Multiscale Model. Simul., 3: 395 - 412, 2005]. An efficient and accurate solver with the ability to resolve these low energy surface fluxes will be an essential ingredient.

With this as our motivation we will describe the development and formulation of a high order solver for a space-time elasto- and visco-dynamic problem formed by merg- ing Hookes law with the Zener and Kelvin-Voigt models for viscoelasticity. We employ a spectral finite element method to discretize in space and a high order discontinuous Galerkin finite element discretization in time using normalized Legendre polynomials of arbitrary degree, r, say. This choice allows the linear system to be decoupled by following Werder et al.s technique [Comp. Meth. Appl. Mech. Engrg., 190: 6685 - 6708, 2001] and results in a set of (r+1) complex symmetric systems for each time slab. We illustrate the effect of the decoupling with respect to accuracy and computation time.

22-4 23: Mini-Symposium: New advances in a posteriori error estimation

23 Mini-Symposium: New advances in a posteriori error estimation

Organisers: Mark Ainsworth, Alexandre Ern and Martin Vohralik

23-1 23: Mini-Symposium: New advances in a posteriori error estimation

COMPUTABLE ERROR BOUNDS FOR FINITE ELEMENT APPROXIMATION ON NON-POLYGONAL DOMAINS Mark Ainsworth1 and Richard Rankin2

1Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island, 02912, USA. Mark [email protected] 2Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, Texas, 77005, USA. [email protected]

We consider the case of piecewise affine finite element approximation of the solution to the Poisson problem with pure Neumann boundary conditions on domains which are non-polygonal. We obtain an a posteriori error estimator which takes the effect of the boundary approximation into account. The estimator provides a guaranteed upper bound on the energy norm of the error and, up to a constant and oscillation terms, local lower bounds on the energy norm of the error.

23-2 23: Mini-Symposium: New advances in a posteriori error estimation

GUARANTEED AND ROBUST ERROR BOUNDS FOR SINGULARLY PERTURBED PROBLEMS IN ARBITRARY DIMENSION Mark Ainsworth1 and Tom´aˇsVejchodsk´y2

1Division of Applied Mathematics, Brown University, Providence, USA, mark [email protected] 2Mathematical Institute, University of Oxford, UK and Institute of Mathematics, Academy of Sciences, Prague, Czech Republic, [email protected]

We present guaranteed, robust, and fully computable error bounds for finite element solutions of singularly perturbed reaction-diffusion problem

−∆u + κ2u = f in Ω, u = 0 in ∂Ω.

The bounds are proved to be robust for the entire range of values of the reaction coeffi- cient κ including the singularly perturbed case. The method is based on equilibration of interelement fluxes and on a subsequent reconstruction of the complementary flux. The construction and equilibration of interelement fluxes that yields robust error bounds was proposed already in [1]. However, error bounds from [1] cannot be computed ex- actly and if they are approximated then they do not guarantee the upper bound on the error. In order to overcome this issue, we applied the complementarity technique in [2] to and we obtained a robust, fully computable and guaranteed upper bounds on the error. Nevertheless, the reconstruction of the complementary flux in [2] is technically demanding and applicable to two-dimensional problems only. In the talk we present a reconstruction of the complementary flux that is more elegant and provides guaranteed, fully computable, and robust error bounds in arbitrary dimension.

References

[1] Ainsworth, M., Babuˇska, I., Reliable and robust a posteriori error estimating for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 331–353.

[2] Ainsworth, M., Vejchodsk´y,T., Fully computable robust a posteriori error bounds for singularly perturbed reaction-diffusion problems. Numer. Math. 119 (2011) 219– 243.

23-3 23: Mini-Symposium: New advances in a posteriori error estimation

INSTANCE OPTIMALITY FOR THE MAXIMUM STRATEGY Lars Diening

LMU Munich, Institute of Mathematics, Theresienstr. 39, 80333 Munich, Germany [email protected]

We study the adaptive finite element approximation of the Dirichlet problem −∆u = f with zero boundary values using linear Ansatz functions and newest vertex bisection. Our approach is based on the minimization of the corresponding Dirichlet energy. We show that the maximums strategy attains every energy level with a number of degrees of freedom, which is proportional to the optimal number. As a consequence we achieve instance optimality of the error. This is a joint work with Christian Kreuzer (Bochum) and Rob Stevenson (Amsterdam).

23-4 23: Mini-Symposium: New advances in a posteriori error estimation

A FRAMEWORK FOR ROBUST A POSTERIORI ERROR CONTROL IN UNSTEADY NONLINEAR ADVECTION-DIFFUSION PROBLEMS V´ıtDolejˇs´ı1, Alexandre Ern2 and Martin Vohral´ık3

1Charles University in Prague, Faculty of Mathematics and Physics, Sokolovsk´a83, 186 75 Praha 8, Czech Republic. [email protected] 2Universit´eParis-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne-la-Vall´ee,France. [email protected] 3INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France. [email protected]

We derive a framework [1] for a posteriori error estimates in unsteady, nonlinear, possi- bly degenerate, advection-diffusion problems. Our estimators are based on a space-time equilibrated flux reconstruction and are locally computable. They are derived for the error measured in a space-time mesh-dependent dual norm stemming from the problem and meshes at hand augmented by a jump seminorm measuring possible nonconformi- ties in space. Owing to this choice, a guaranteed and globally efficient upper bound is achieved, as well as robustness with respect to nonlinearities, advection dominance, domain size, final time, and absolute and relative size of space and time steps. Local- in-time and in-space efficiency is also shown for a localized upper bound of the error measure. In order to apply the framework to a given numerical method, two simple conditions, local space-time mass conservation and an approximation property of the reconstructed fluxes, need to be verified. We show how to do this for the interior- penalty discontinuous Galerkin method in space and the Crank–Nicolson scheme in time. Numerical experiments illustrate the theory.

References

[1] Dolejˇs´ıV., Ern A., Vohral´ık,M., A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems. SIAM J. Numer. Anal. (2013), DOI 10.1137/110859282.

23-5 23: Mini-Symposium: New advances in a posteriori error estimation

QUASI-OPTIMAL AFEM FOR NON-SYMMETRIC OPERATORS Michael Feischla Thomas F¨uhrerb and Dirk Praetoriusc

Vienna University of Technology, Institute for Analysis and Scientific Computing, Vienna, Austria [email protected], [email protected], [email protected]

In our talk, we present our recent preprint [arXiv:1210.8369], where adaptive mesh- refinement for conforming FEM of general linear, elliptic, second-order PDEs is ana- lyzed. For a bounded Lipschitz domain Ω ⊂ Rd, our model problem thus reads Lu(x) := −divA(x)∇u(x) + b(x) · ∇u(x) + c(x)u(x) = f(x) x ∈ Ω, u(x) = 0 x ∈ ∂Ω.

For a given conforming simplicial mesh T`, we allow continuous T`-piecewise polyno- mials of arbitrary, but fixed polynomial order with homogeneous boundary conditions p S0 (T`) as ansatz functions. As e.g. in [Cascon-Kreuzer-Nochetto-Siebert, SINUM 2008], adaptivity is driven by the residual error estimator ρ`, and we prove convergence even with quasi-optimal algebraic convergence rates. The advantages over the state of the art read as follows: Unlike prior works for linear non-symmetric operators, e.g. [Cascon-Nochetto, IMA JNA 2012], our analysis avoids the artificial quasi-symmetry assumptions ∇ · b = 0 and c ≥ 0 as well as the interior node property for the refinement. Moreover, the differential operator L has to satisfy a G˚ardinginequality only. If L is uniformly elliptic, no additional assumption on the initial mesh is posed. Finally, our analysis also covers certain nonlinear problems in the frame of strongly monotone operators. On a technical level, the heart of the matter is a novel quasi-orthogonality estimate which builds on the observation that estimator reduction already implies convergence of the adaptive scheme. Moreover and unlike e.g. [Cascon-Kreuzer-Nochetto-Siebert, SINUM 2008] and [Cascon-Nochetto, IMA JNA 2012], our analysis avoids the use of the so-called total error and quasi error and directly works with the estimator. As a consequence, the bound for the range of optimal marking parameters 0 < θ < θ? does not depend on lower bounds for the error (so-called efficiency estimates).

23-6 23: Mini-Symposium: New advances in a posteriori error estimation

A POSTERIORI ERROR ESTIMATES FOR THE WAVE EQUATION Omar Lakkis1, Emmanuil H. Georgoulis2 and Charalambos Makridakis3

1Department of Mathematics, University of Sussex, Falmer near Brighton, England UK [email protected] 2Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom 3Department of Applied Mathematics, University of Crete, L. Knosou GR 71409 Heraklion, Greece

We present a posteriori error bounds in L∞(0,T,L2(Ω)) norm for an implicit time- stepping method in combination with a finite element method in space to solve the initial-boundary value problem for the wave equation on the space-time domain Ω × (0,T ). The technique relies on appropriately constructed time-reconstruction in com- bination with Baker’s function method from Baker (SINUM, 1976). I will mention related results concerning the leapfrog method as well. In the latter case, the time- reconstruction has to be also carefully designed as to obtain optimal-order estimators.

23-7 23: Mini-Symposium: New advances in a posteriori error estimation

ON MATHEMATICAL METHODS GENERATING FULLY RELIABLE A POSTERIORI ESTIMATES FOR NONLINEAR BOUNDARY VALUE PROBLEMS Sergey Repin

V. A. Steklov Institute of Mathematics at St. Petersburg, Fontanka 27, 191023, St. Petersburg, Russian Federation [email protected]

Modern theory of partial differential equations has elaborated several ways of deriving fully guaranteed estimates, which can be used as a posteriori estimates for numerical (e.g., FEM) solutions and as computable bounds of modeling errors. For problems generated by linear differential operators, the corresponding theory is well developed. It is difficult to say the same about many classes of nonlinear problems, which are much less studied in the context of a posteriori analysis. One of the most important questions is how to define the right (suitable) measure of the error for a strongly nonlinear problem. We discuss possible answers to this and other questions with the paradigm of a class of convex variational problems and motivate the selection of a certain error measure. It is supplied with bounds, which are directly computable (i.e., they do not require exact satisfaction of some additional conditions). Also, we discuss closely related mathematical questions, which are principally important for quantitative analysis of nonlinear problems and present generalized forms of the Prager- Synge estimate, Mikhlin’s variational identity, Helmgholtz decomposition theorem, and derive a general estimate of the distance to the set equilibrated fields.

ADAPTIVE FINITE ELEMENTS FOR PDE CONSTRAINED OPTIMAL CONTROL PROBLEMS Kunibert G. Siebert

Institute for Applied Analysis and Numerical Simulation, Universit¨atStuttgart, Germany [email protected]

Many optimization processes in science and engineering lead to optimal control prob- lems where the sought state is a solution of a partial differential equation (PDE). Control and state may be subject to further constraints. The complexity of such prob- lems requires sophisticated techniques for an efficient numerical approximation of the true solution. One particular method are adaptive finite element discretizations. We report on ongoing research about control constrained optimal control problems. We give a summary about recent findings concerning sensitivity analysis, a posteriori error control, and convergence of adaptive finite elements. This is joint work with Fernando D. Gaspoz (Stuttgart).

23-8 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

24 Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

Organisers: Axel Klawonn and Gerhard Starke

24-1 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

A NEW COARSE SPACE FOR FETI-DP IN THE CONTEXT OF ALMOST INCOMPRESSIBLE ELASTICITY Sabrina Gippert1a. Axel Klawonn 1b and Oliver Rheinbach2

1Mathematisches Institut, Universit¨atzu K¨oln, Germany [email protected], [email protected] 2Fakult¨atf¨urMathematik und Informatik, Technische Universit¨atBergakademie Freiberg, Germany [email protected]

Domain decomposition methods such as FETI-DP have been considered successfully for almost incompressible elasticity problems. To ensure a good condition number it is known, that for mixed finite element discretizations with discontinuous pressure elements a zero net flux condition on each subdomain is needed. This is usually done en- forcing the constraint for each edge and each face separately. Here, the edge constraints are implemented using a transformation of basis approach with partial assembly, while the face terms are enforced using a deflation framework. For this new approach it is sufficient to implement the zero net flux condition in this deflation method using one constraint for each subdomain instead of one constraint for each face, which allows a much smaller coarse space. This new approach will be discussed and numerical results will be presented.

NONLINEAR FETI-DP AND BDDC METHODS Axel Klawonn1a, Martin Lanser1b, and Oliver Rheinbach2

1Mathematisches Institut, Universit¨atzu K¨oln, Germany [email protected], [email protected] c Fakult¨atf¨urMathematik und Informatik, Technische Universi¨atBergakademie Freiberg, [email protected]

New nonlinear FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods are introduced. In all of these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches have a potential to reduce communication and show a significantly improved performance. Numerical results for the p-Laplace operator are presented.

24-2 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

LSFEM FOR GEOMETRICALLY AND PHYSICALLY NONLINEAR ELASTICITY PROBLEMS Benjamin M¨uller1a, Gerhard Starke1b, J¨orgSchr¨oder2c, Alexander Schwarz2d and Karl Steeger2e

1Faculty of Mathematics, University Duisburg - Essen, Essen, Germany [email protected], [email protected] 2Faculty of Engineering, University Duisburg - Essen, Essen, Germany [email protected], [email protected] [email protected]

Deformation processes of solid materials are omnipresent and can be described by sys- tems of partial differential equations in continuum mechanics. In this talk we present a least squares finite element method based on the momentum balance and the constitu- tive equation for hyperelastic materials. Our approach is motivated by a well - studied least squares formulation for linear elasticity. This method is generalized to an ap- proach which takes physical as well as geometrical nonlinearities into account. The novelty of our approach is that, in addition to the displacement u, we consider the full first Piola - Kirchhoff stress tensor P and approximate both simultaneously. In the discrete formulation we use quadratic Raviart - Thomas elements for the stress tensor and continuous quadratic elements for the displacement vector. For the min- imization of the nonlinear least squares functional, the Gauss - Newton method with backtracking line search is used. We will emphasize the advantages of our approach in comparison to other solution methods. The talk will end with an illustration of the performance for some two di- mensional problems in plane strain configuration and some three dimensional problems.

AN APPROACH TO ADAPTIVE COARSE SPACES IN FETI-DP METHODS Oliver Rheinbach1, Axel Klawonn2b and Patrick Radtke2c

1 Fakult¨atf¨urMathematik und Informatik, Technische Universit¨atBergakademie Freiberg, Germany [email protected] 2 Mathematisches Institut, Universit¨atzu K¨oln, Germany [email protected], [email protected]

Adaptive coarse spaces for domain decomposition methods can be used to obtain inde- pendence on coefficient jumps for highly heterogeneous problems, even when coefficient jumps inside subdomains are present. In this talk, for FETI-DP methods, we present a new approach to obtain independence of the coefficient jumps by solving certain local eigenvalue problems and enriching the FETI-DP coarse space with eigenvectors.

24-3 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

GEODESIC FINITE ELEMENTS Oliver Sander

Institut f¨urGeometrie und Praktische Mathematik, RWTH Aachen University, Germany [email protected]

Geodesic finite elements are a novel way to discretize problems involving functions with values on a Riemannian manifold. Examples for such problems include Cosserat mate- rials and liquid crystals. Geodesic finite elements of any order can be constructed, and are conforming in the sense that they are first-order Sobolev functions. The construc- tion is equivariant under isometries of the value manifold, which means that frame- indifference in mechanics is preserved. Optimal discretization error bounds have been shown analytically, and can be observed in numerical experiments. We present the theory of geodesic finite elements and give a few example applications.

24-4 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

ASPECTS ON MIXED LEAST-SQUARES FINITE ELEMENTS FOR HYPERELASTIC PROBLEMS Alexander Schwarz1a, Karl Steeger1ba, J¨orgSchr¨oder1c, Gerhard Starke2d and Benjamin M¨uller2e

1 Faculty of Engineering, Universit¨atDuisburg-Essen, Germany [email protected], [email protected], [email protected] 2 Faculty of Mathematics, Universit¨atDuisburg-Essen, Germany [email protected], [email protected]

The main goal of this contribution is the solution of geometrically nonlinear prob- lems using the mixed least-squares finite element method (LSFEM). The basis for the proposed element formulations are div-grad first-order systems consisting of the equi- librium condition and the constitutive equation both written in residual forms, see e.g. [2] and [1]. Generally, L2-norms are adopted on these residuals leading to function- als depending on displacements and stresses, which are the basis for the associated minimization problems. In particular we consider different hyperelastic free energy functions in order to define the stress response of the material. Besides some numer- ical advantages, as e.g. an inherent symmetric structure of the system of equations and an error estimator, it is known that least-squares methods have also a drawback concerning accuracy, especially when lower-order elements are used. Therefore, a focus of the presentation is on performance and implementation aspects of triangular mixed finite elements with different interpolation order. In order to approximate the stresses, shape functions related to the edges are chosen. These vector-valued functions are used for the interpolation of the rows of the stress tensor and belong to a Raviart-Thomas space, which guarantees a conforming discretization of the Sobolev space H(div). Fur- thermore, standard polynomials associated to the vertices of the triangle are used for the continuous approximation of the displacements in W 1,p with p > 2. Finally, the proposed formulations will be compared considering various benchmark problems, computational costs and accuracy.

References

[1] A. Schwarz, J. Schr¨oder, G. Starke, and K. Steeger. Least-squares mixed finite elements for hyperelastic material models. In Report of the Workshop 1207 at the “Mathematisches Forschungsinstitut Oberwolfach” entitled “Advanced Compu- tational Engineering”, organized by O. Allix, C. Carstensen, J. Schr¨oder,P. Wrig- gers, pages 14–16, 2012.

[2] G. Starke, B. M¨uller,A. Schwarz, and J. Schr¨oder.Stress-displacement least squares mixed finite element approximation for hyperelastic materials. In Report of the Workshop 1207 at the “Mathematisches Forschungsinstitut Oberwolfach” entitled “Advanced Computational Engineering”, organized by O. Allix, C. Carstensen, J. Schr¨oder,P. Wriggers, pages 11–13, 2012.

24-5 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

ON ISOGEOMETRIC FINITE ELEMENTS IN SOLID MECHANICS AND VIBRATIONAL ANALYSIS Bernd Simeona and Oliver Weegerb

Department of Mathematics, Felix-Klein-Zentrum, TU Kaiserslautern, Germany. [email protected], [email protected]

Over the last years, the new paradigm of Isogeometric Analysis has demonstrated its potential to bridge the gap between Computer Aided Design (CAD) and the Finite Element Method (FEM). The distinguished feature of isogeometric finite elements is the usage of one common geometry representation for creating CAD models, for meshing, and for numerical simulation. In this way, a seamless integration of all computational tools within a single design loop comes into reach. Moreover, increased smoothness of the basis functions and an exact representation of the boundary are properties which are also attractive from a numerical viewpoint. The presentation is aimed at the application of isogeometric methods in the field of solid mechanics, in particular vibrational analysis. We start with a short overview on the methodology, point out the common features and differences when compared to the classical finite element method, and concentrate then on static and vibrational analysis of linear and nonlinear elasticity problems. By studying typical geometries and problem setups, we analyze the pros and cons of employing NURBS-based shape functions. In this context, we also summarize our work on implementing a 3D isogeometric solver with multi-patch and large deformation capabilities. Finally, we explore the benefits of higher smoothness in the field of non- linear vibrational analysis. The method of harmonic balance is well-established here and allows the tracing of resonance phenomena. Using a well-understood nonlinear Euler-Bernoulli beam model as benchmark problem, we demonstrate that spline-based discretizations with higher global smoothness lead to an improved accuracy in the harmonic balance analysis and are thus a promising approach for this particular appli- cation. This work is supported by the European Union within the FP7-project TERRIFIC: Towards Enhanced Integration of Design and Production in the Factory of the Future through Isogeometric Technologies.

24-6 24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

MOMENTUM BALANCE IN FIRST-ORDER SYSTEM FINITE ELEMENT METHODS FOR ELASTICITY Gerhard Starke

University Duisburg-Essen, Germany [email protected]

The purpose of this talk is to demonstrate the importance of accurate momentum bal- ance approximation for stress-displacement first-order system formulations in elasticity. Our study includes first-order system least squares methods as well as mixed methods of saddle point structure and establishes close relations between these different ap- proaches. Since the latter class of methods treat momentum balance as a constraint, this approach appears to be favourable. With a slight modification, however, first-order system least squares formulations also lead to surprisingly high momentum balance ac- curacy. The advantage of this approach is that Raviart-Thomas elements (of degree k) for the stress approximation can be combined with standard conforming elements (of degree k +1) for the displacements without stability problems. For bending-dominated problems it is observed that improved momentum balance accuracy goes hand in hand with much better convergence of pointwise values. While the theoretical connections are mainly derived for linear elasticity models we also investigate geometrically non- linear elasticity using a hyperelastic material of Neo-Hookean type.

24-7 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

25 Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

Organisers: Antti Hannukainen and Lothar Nannen

25-1 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

ANALYSIS OF A CARTESIAN PML APPROXIMATION TO ACOUSTIC SCATTERING PROBLEMS IN RN James H. Bramblea and Joseph E. Pasciakb

Department of Mathematics, Texas A&M University, College Station, TX, USA. [email protected], [email protected]

We consider the application of a perfectly matched layer (PML) technique in Cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain. This work provides new stability estimates for both the infinite domain PML approximation as well as the truncated (bounded) domain PML approximation. The stability estimates are new in the sense that they demonstrate the interplay between the size of the computational domain M and the strength of the PML stretching function σ0. In particular, we show that the stability and exponential convergence depends on the product σ0M. This means that one can obtain the desired accuracy without changing the domain size by simply increasing the PML parameter σ0. We illustrate these results with computations showing that even infinitesimal PML layers are adequate provided that the PML strength is suitably adjusted. These stability estimates allow for the case of piecewise C1 stretching as well as piecwise C0 stretching. The second case gives rise to PML variational equations with jumping coefficients.

25-2 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

A HIGH FREQUENCY BEM FOR SCATTERING BY A CLASS OF NONCONVEX OBSTACLES Simon Chandler-Wilde1, David Hewett1a,2, Stephen Langdon1 and Ashley Twigger1

1Department of Mathematics and Statistics, University of Reading, UK. [email protected] 2Current address: Mathematical Institute, University of Oxford, UK.

There is considerable current interest in the development of numerical methods for time-harmonic acoustic and electromagnetic scattering problems that are able to effi- ciently resolve the scattered field at high frequencies. Conventional finite or boundary element methods, with piecewise polynomial approximation spaces, suffer from the re- striction that a fixed number of degrees of freedom is required per wavelength in order to represent the oscillatory solution, leading to excessive computational cost when the scatterer is large compared to the wavelength. The ‘hybrid numerical-asymptotic’ (HNA) approach aims to reduce the computa- tional cost at high frequencies by using a numerical approximation space incorporating oscillatory functions, chosen based on partial knowledge of the high frequency asymp- totic behaviour of the solution. This is a fast-evolving field - for a recent review of the HNA approach in the boundary element context, see [1]. However, to date, the vast majority of algorithms, and all the numerical analysis, have been restricted to problems of scattering by single convex obstacles (e.g. [2]). In this talk we describe a HNA hp-BEM for acoustic scattering by a class of sound- soft nonconvex polygons. We demonstrate via a rigorous, frequency-explicit error anal- ysis, supported by numerical examples, that to achieve any desired accuracy it is suffi- cient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. The main difficulty in moving from the convex case to the nonconvex case is that the high frequency asymptotic behaviour of the solution, knowledge of which is required for the design of the HNA approximation space, becomes significantly more complicated. In particular, there are two new complexities to consider: first, multiply-reflected and diffracted-reflected rays may be present in the asymptotic solution; second, one must deal with the rapid variation of the field across the shadow boundaries between the illuminated and shadow regions of previously diffracted ray fields. Our full hp-error analysis is based on new results concerning the high frequency asymptotic solution, and its analytic continuation into the complex plane. Further details are available in [3].

References

[1] S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scat-

25-3 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

tering, Acta Numer., 21 (2012), pp. 89–305. [2] D. P. Hewett, S. Langdon and J. M. Melenk, A high frequency hp boundary element method for scattering by convex polygons, SIAM J. Numer. Anal., 51(1) (2013), pp. 629–653. [3] S. N. Chandler-Wilde, D. P. Hewett, S. Langdon, and A. Twigger, A high frequency boundary element method for scattering by a class of noncon- vex obstacles , University of Reading Department of Mathematics and Statistics preprint MPS-2012-04, submitted for publication.

A HIGH FREQUENCY BOUNDARY ELEMENT METHOD FOR SCATTERING BY TWO-DIMENSIONAL SCREENS Simon Chandler-Wilde1, David Hewett1,2, Stephen Langdon1c and Ashley Twigger1

1Department of Mathematics and Statistics, University of Reading, UK. [email protected] 2Current address: Mathematical Institute, University of Oxford, UK.

We propose a numerical-asymptotic boundary element method for problems of time- harmonic acoustic scattering of an incident plane wave by a sound-soft two-dimensional screen. Standard numerical schemes have a computational cost that grows at least linearly with respect to the frequency of the incident wave. Here, we enrich our ap- proximation space with oscillatory basis functions carefully designed to capture the high frequency behaviour of the solution. We show that in order to achieve any desired accuracy it is sufficient to increase the number of degrees of freedom only in proportion to the logarithm of the frequency, as the frequency increases, and for fixed frequency we demonstrate exponential convergence with respect to the number of degrees of freedom.

25-4 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

SOLVING THE STEADY-STATE AB-INITIO LASER THEORY WITH FEM Sofi Esterhazy1, Matthias Liertzer2a, Jens Markus Melenk1 and Stefan Rotter2

1Institute for Analysis and Scientific Computing, Vienna University of Technology, A-1040 Vienna, Austria, EU 2Institute for Theoretical Physics, Vienna University of Technology, A-1040 Vienna, Austria, EU [email protected]

The key equations of semi-classical laser theory are the Maxwell-Bloch equations. This time-dependent set of nonlinearly coupled partial differential equations describes a wealth of phenomena in laser physics ranging from chaotic or pulsed lasing to the most commonly studied scenario of steady-state lasing. In the latter case the laser output consists of several harmonically oscillating modes which can be described by a reduced set of time-independent nonlinearly coupled Helmholtz equations with open boundary conditions [1]. In this only very recently proposed approach, which is known as the steady-state ab-initio laser theory (SALT), each laser mode Ψµ is described as follows, " !# 2 γ⊥ D0(x) ∆ + k εc(x) + Ψµ(x) = 0, (1) µ k − k + iγ PN 2 µ a ⊥ 1 + ν=1 Γν|ψν(x)| where all of these N equations (i.e., µ = 1 ...N) are nonlinearly coupled to each other PN 2 by the denominator ν=1 Γν|ψν(x)| . Note that both the number of modes N, as well as the modes Ψµ and the corresponding frequencies kµ are a priori unknown and need to be found within the constraint that the frequencies kµ are real. The given parameters are the dielectric function εc(x), the Lorentzian gain curve with a peak frequency of ka and width 2γ⊥, and the pump D0(x). Up to now, the above SALT equations have only been solved by expanding the modes Ψµ on a special set of biorthogonal basis vectors, also known as constant-flux states [1]. In this talk I will present a novel strategy for efficiently solving these equa- tions on top of a finite element discretization without the requirement of such a basis set. For this purpose we employ a nonlinear eigensolver for treating the frequencies kµ, as well as an iterative Newton-Raphson scheme for dealing with the nonlinearity in Ψµ.

References

[1] H.E. T¨ureci, A.D. Stone, B. Collier, Phys. Rev. A 74, 043822; H.E. T¨ureci, A.D. Stone, L. Ge, S. Rotter, R.J. Tandy, Nonlinearity 22, C1 (2009), L. Ge, Y. Chong, A.D. Stone, Phys. Rev. A 82, 063824 (2010)

[2] S. Esterhazy, M. Liertzer, J.M. Melenk, S. Rotter, in preparation

25-5 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

HOW SHOULD ONE CHOOSE THE SHIFT FOR THE SHIFTED LAPLACIAN TO BE A GOOD PRECONDITIONER FOR THE HELMHOLTZ EQUATION? Martin Gander1, I. G. Graham2b and E. A. Spence2c

1 Section de Math´ematiques,Universit´ede Gen`eve, CH-1211 Gen`eve, Switzerland, [email protected] 2 Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK, [email protected], [email protected]

There has been much recent research on preconditioning the Helmholtz equation ∆u + k2u = 0 with the inverse of the operator ∆+(k2+iε) (i.e. the Helmholtz operator with a complex shift), with this method often known as “the shifted Laplacian preconditioner”. Despite many numerical investigations there has been relatively little analysis of how one should chose the shift, ε, for the type of Helmholtz problems arising in applications. In this talk we present sufficient conditions for the shifted problem to be a good preconditioner for the original Helmholtz problem for finite element discretisations of the following Helmholtz boundary value problems: (i) the interior impedance problem, and (ii) the sound-soft scattering problem (with the radiation condition imposed as an impedance boundary condition). For example, let Ω be a bounded Lipschitz domain in Rd with boundary Γ. Let A denote the matrix arising when the standard variational formulation of the problem ∂u ∆u + k2u = −f in Ω, − iku = g on Γ, (1) ∂n is solved using the Galerkin method, and let Aε denote the Galerkin matrix arising from the standard variational formulation of the problem with ε, i.e. ∂u ∆u + (k2 + iε)u = −f in Ω, − iku = g on Γ. (2) ∂n

−1 Theorem (Sufficient conditions for Aε to be a good preconditioner for A) Suppose that either Ω is a C1,1 domain in 2- or 3-d that is star-shaped in the sense that in the sense that inf (x · n(x)) > 0, x∈Γ or Ω is a convex polygon or polyhedron, and suppose that the Galerkin discretisations of both the Helmholtz problem (1) and the problem with a shift (2) are formed using conforming, piecewise-linear finite elements on a quasi-uniform mesh. Assume that there exists a c > 0 such that ε ≤ ck2 for all k. Then, there exists a p 2 k0 > 0 and C1,C2 > 0 (independent of h, k, and ε) such that if hk |k − ε| ≤ C1 then

−1 ε I − A A ≤ C2 ε 2 k for all k ≥ k0. −1 Thus, if ε/k is sufficiently small, Aε is a good preconditioner for Aε.

25-6 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

HYBRID NUMERICAL-ASYMPTOTIC APPROXIMATION FOR HIGH FREQUENCY SCATTERING BY PENETRABLE CONVEX POLYGONS Samuel Grotha, David Hewett and Stephen Langdon

Department of Mathematics and Statistics, University of Reading, UK. [email protected]

In this talk we consider the two-dimensional problem of scattering of a time-harmonic wave by a penetrable convex polygon Ω with boundary ∂Ω. Standard numerical meth- ods for scattering problems, using piecewise polynomial approximation spaces, require a fixed number of degrees of freedom per wavelength in order to represent the os- cillatory solution. This leads to prohibitive computational expense in the high fre- quency regime. For problems of scattering by impenetrable scatterers, where there is just one wavenumber k, much work has been done on developing and analysing hybrid numerical-asymptotic (HNA) methods (see [1]) which overcome this limitation. These HNA methods approximate the unknown boundary data v in a boundary element method framework using an ansatz of the form

M X v(x, k) ≈ v0(x, k) + vm(x, k) exp(ikψm(x)), x ∈ ∂Ω, (1) m=1 where the phases ψm are chosen using knowledge of the high frequency asymptotics. The expectation is that if v0 (the geometric optics) and ψm are chosen well, then vm will be much less oscillatory than v and so can be more efficiently approximated by piecewise polynomials than v itself. This talk discusses the challenging task of generalising the HNA methodology to so-called “transmission problems” involving penetrable scatterers. The main difficulty in this generalisation is that the high frequency asymptotic behaviour is significantly more complicated than for the impenetrable case. In particular, the boundary of the scatterer represents an interface between two media with different wavenumbers, and so we expect to need to modify the ansatz (1) to include terms oscillating at both wavenumbers. We discuss how appropriate phases are chosen in the penetrable case using high frequency asymptotics and hence show how effective HNA approximation spaces can be constructed for this problem. Moreover, we demonstrate, via comparison with a reference solution, that these HNA approximation spaces can approximate the highly oscillatory solution of the transmission problem accurately and efficiently at all fre- quencies. Full details can be found in [2].

References

[1] S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scat- tering, Acta Numer., 21 (2012), pp. 89–305.

25-7 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

[2] S. P. Groth, D. P. Hewett, S. Langdon, Hybrid numerical-asymptotic ap- proximation for high frequency scattering by penetrable convex polygons , Univer- sity of Reading Department of Mathematics and Statistics preprint MPS-2013-02, submitted for publication.

25-8 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

ANALYSIS OF PRECONDITONERS FOR HELMHOLTZ EQUATION USING PESUDOSPECTRUM Antti Hannukainen

Department of Mathematics and Systems Analysis, Aalto University, Finland. [email protected]

Finite element simulation of time-harmonic wave propagation problems leads to so- lution of very large indefinite linear systems. When losses, absorbing boundary or impedance boundary conditions are present, as often in realistic engineering applica- tions, these linear systems are complex valued, non-Hermitian and non-normal. Solving such systems is very challenging. The large size of the system restricts the use of di- rect solvers making preconditioned iterative solvers the method of choice, especially in high-frequency domain. Analyzing the convergence properties of such preconditioned iterative methods for wave-propagation problems is difficult. This is mainly due to indefiniteness and non- normality. Because of the non-normality, the eigenvalues alone do not give information of the convergence. This problem can be handled by using a suitable convergence criterion. In this talk, we focus on convergence analysis of the preconditioned GMRES method for the Helmholtz equation with first order absorbing boundary conditions. For GM- RES, convergence criteria suitable for analyzing non-normal systems are based on es- timating the field of values (FOV) or the pseudospectrum. The FOV based convergence criterion has been used to study two-level and Laplace preconditioners for Helmholtz equation in media with losses, [1, 2]. The major short- coming of this approach is in handing indefiniteness. This is due to the fact that the FOV is always a convex set containing all eigenvalues. The FOV delivers GMRES con- vergence estimates only when the origin does not belong to the FOV, thus it cannot be successfully applied when the eigenvalues are clustered around the origin, although they would be far from it. This is the case for time-harmonic Helmholtz equation with absorbing boundary conditions. Due to this difficulty, we consider a pseudospectrum based convergence criteria. In this talk, we demonstrate how it can be applied to analyze preconditioners when the FOV based criteria fails.

References

[1] A. Hannukainen. Field of values analysis of a two-level preconditioner for the Helmholtz equation. SIAM Journal on Numerical Analysis, (accepted for publica- tion), 2013.

[2] M. B. van Gijzen, Y. A. Erlangga, and C. Vuik. Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput., 29(5):1942–1958, 2007.

25-9 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

A DOMAIN DECOMPOSITION PRECONDITIONER FOR MIXED HYBRID INFINITE ELEMENTS Martin Hubera, Lothar Nannenb and Joachim Sch¨oberlc

Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria [email protected], [email protected], [email protected]

In a direct numerical simulation of scalar time-harmonic scattering problems the num- ber of unknowns typically increase like O(κ3) with increasing wavenumber κ. Due to the pollution effect the situation is even worse. Since direct solvers like PARDISO or MUMPS are due to the memory requirements only useful for small and medium scale problems, iterative solvers are of interest for large scale problems. Because usual finite element methods lead to indefinite discretization matrices, a suitable preconditioner is needed to ensure convergence of the iterative solver. An additional difficulty arises from the fact, that scattering problems are non-local and therefore often a transpar- ent boundary condition is needed to restrict the problem to a bounded computational domain. In a standard variational formulation of the Helmholtz equation −∆u − κ2u = f the solution u is assumed to be in H1(Ω). In this talk we are using the mixed hybrid formulation of [1] for the solution u ∈ L2(Ω), the discontinuous gradient field σ := 1 2 iκ ∇u ∈ Hdiscont(div, Ω), the solution uF ∈ L (F) on the skeleton F of a triangulation 2 T and the normal components of the gradient field σF ∈ L (F) on the skeleton. Since the volume terms are local, they can be eliminated with static condensation and a discrete problem for the skeleton variables uF and σF remains. We use a Robin-type domain decomposition preconditioner with additional penalty terms in the mixed hybrid formulation, which was presented by Huber and Sch¨oberl on the international conference on domain decomposition methods 2012 in Rennes (see [2]). This preconditioner does not need a coarse grid correction and is therefore well suited for large scale problems. To realize the transparent boundary condition the method is completed by a mixed hybrid discontinuous version of the Hardy space infinite elements presented in [3].

References

[1] P. Monk, J. Sch¨oberl, A. Sinwel. Hybridizing Raviart-Thomas elements for the Helmholtz equation. Electromagnetics, 30(1): 149–176, 2010.

[2] M. Huber. Hybrid discontinuous Galerkin methods for the wave equation. PhD thesis, Vienna University of Technology, Austria, 2013.

[3] L. Nannen, T. Hohage, A. Sch¨adle, and J. Sch¨oberl. Exact sequences of high order Hardy space infinite elements for exterior Maxwell problems. SIAM J. Sci. Comput., 2013, published online: http://dx.doi.org/10.1137/110860148.

25-10 25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

IMPROVING THE SHIFTED LAPLACE PRECONDITIONER BY MULTIGRID DEFLATION A. H. Sheikha, D. Lahayeb and C. Vuikc

Delft Institute of Applied Mathematics, TU Delft, Delft, NL. [email protected], [email protected], [email protected]

How does deflation by multigrid vectors affect the performance of the shifted Lapla- cian preconditioned for the Helmholtz equation? We investigate two deflation variants that differ in the choice of the coarse grid operator. A rigorous Fourier mode analysis for the one-dimensional problem with Dirichlet boundary conditions shows that the use of deflation results in tighter clustering of the spectrum at low wavenumber, and that undesirable small eigenvalues reappear at high wavenumber. Numerical results for two-dimensional problems show an iteration count that remains constant for low wavenumber and that increases linearly after a certain threshold value. This thresh- old value is larger in the deflation variant with the coarse grid operator that is more expensive to compute. Numerical results with the multilevel extension of the defla- tion algorithm on three-dimensional problems shows a speed-up for sufficiently large problems size.

Reference A. H. Sheikh, D. Lahaye and C. Vuik, On the convergence of shifted Laplace preconditioner combined with multilevel deflation, NLAA, 2013 (DOI: 10.1002/nla.1882).

25-11 26: Mini-Symposium: Numerical Methods for Parabolic Equations

26 Mini-Symposium: Numerical Methods for Parabolic Equations

Organiser: Thomas Wihler

26-1 26: Mini-Symposium: Numerical Methods for Parabolic Equations

ENERGY CONSERVATIVE/DISSIPATIVE APPROXIMATIONS OF NONLINEAR EVOLUTION PROBLEMS Charalambos Makridakis

School of Mathematical and Physical Sciences, University of Sussex, UK. and Department of Applied Mathematics, University of Crete, Greece [email protected]

We discuss DG methods for nonlinear evolution PDEs including and higher-order terms describing diffusion and dispersion/capillarity effects. In particular we shall consider the isothermal Navier-Stokes Korteweg system for which we present thermodynami- cally consistent DG schemes. We discuss issues related to the error analysis of the approximations by utilizing appropriate local reconstructions.

26-2 26: Mini-Symposium: Numerical Methods for Parabolic Equations

A POSTERIORI ERROR ANALYSIS FOR DG IN TIME ALE FORMULATIONS Andrea Bonito1, Irene Kyza2 and Ricardo H. Nochetto3

1Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA, [email protected], 2Division of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK & IACM-FORTH, Nikolaou Plastira 100, Vassilika Vouton, GR 700 13 Heraklion-Crete, Greece, [email protected], 3Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA, [email protected]

Arbitrary Lagrangian Eulerian (ALE) formulations are useful when approximating so- lutions of problems defined in time-dependent domains, such as fluid-structure inter- actions. For realistic simulations involving fluids in 3d, it is important that the ALE method is at least of second order of accuracy. For finite element (FE) schemes, higher order in time ALE formulations, without any constraints on the time-steps, were not available in the literature before [1, 2]. In [1], we propose unconditionally stable discon- tinuous Galerkin (dG) methods in time of any order, for a time-dependent diffusion- dominated problem defined on deformable domains, by enforcing a discrete Reynolds’ identity. In [2], we provide optimal order a priori error estimates. As a continuation to [1, 2], in this talk, we discuss a posteriori error analysis for the dG methods introduced in [1]. The analysis is based on the reconstruction technique, and in particular, in the proposition of a novel dG reconstruction in the ALE frame- work. This reconstruction generalises the notion of the dG reconstruction for problems on time-independent domains, introduced earlier by Makridakis & Nochetto in [3]. Us- ing the properties of the reconstruction and pde techniques for the problem written on the ALE framework, we prove optimal order a posteriori error bounds without any restrictions on the time-steps. Our analysis allows variable time-steps and gives im- portant information on the behaviour of the error with respect to the movement of the domain. More precisely, adaptivity is proven to be beneficial in cases of highly oscillatory ALE maps. The analysis is illustrated by insightful numerical experiments.

References

[1] A. Bonito, I. Kyza, R.H. Nochetto, Time discrete higher order ALE formulations: Stability, SIAM J. Numer. Anal. 51 (2013) 577–604. [2] A. Bonito, I. Kyza, R.H. Nochetto, Time discrete higher order ALE formulations: A priori error analysis, to appear in Numer. Math., DOI: 10.1007/s00211-013- 0539-3.

26-3 26: Mini-Symposium: Numerical Methods for Parabolic Equations

[3] Ch. Makridakis, R.H. Nochetto, A posteriori error analysis for higher order dis- sipative methods for evolution problems,Numer. Math. 104 (2006) 489–514.

DISCONTINUOUS GALERKIN APPROXIMATION OF POROUS FISHER-KOLMOGOROV EQUATIONS Fausto Cavalli1a, Giovanni Naldi1b and Ilaria Perugia2

1Dipartimento di Matematica, Universit`adi Milano, Via Saldini 50, 20133 Milano, Italy [email protected], [email protected] 2Dipartimento di Matematica, Universit`adi Pavia, Via Ferrata 1, 27100 Pavia, Italy [email protected]

The reaction-diffusion equation plays an important role in dissipative dynamical sys- tems for physical, chemical and biological phenomena. For example, in chemical physics, in order to describe concentration and temperature distributions, the heat and mass transfer are modeled by a non linear diffusion term, while the rate of heat and mass production are described by a non linear reaction term. These terms are often considered in the form of mass action law. In population dynamics, where the focus is on the evolution of a population density, diffusion terms correspond to a random motion of individuals, and reaction terms describe their reproduction and interaction, as in a predator-prey model. Originally, continuum models of population spreading were based on linear diffusion. Then, several authors have pointed out that these diffusion mechanisms can be more realistically described by (degenerate) non linear diffusion models. The typical non linear reaction-diffusion model is as follows: ∂u − ∆p(u) = r(u) in Ω × (0, +∞), ∂t ∇p(u) · nΩ = 0 on ∂Ω × (0, +∞),

u|{t=0} = u0 in Ω. where Ω ⊂ Rd, with d = 1, 2, is a bounded domain, u is a density, p(u) is a suitable γ diffusivity, r(u) a reaction function, and u0 the initial datum. If p(u) ' u , with γ > 0, the previous equation is known as the porous media equation, which is degenerate in the sense that p0(0) = 0. In this case the solution can develop interfaces between the regions with zero and non zero population density. These sharp fronted solutions represent population density profiles. The reaction is modeled by a generalized Fisher- Kolmogorov term, namely r(u) = u(1 − uβ). From a numerical point of view, the discretization of this problem is challenging because the numerical scheme has to reproduce shock waves or fronts of the analytical solutions, and preserve stability and invariance properties. We present a new high-order numerical methods for the approximation of the porous Fischer-Kolmogorov equation, based on a discontinuous Galerkin space discretization and Runge-Kutta time stepping. These methods are capable to reproduce the main properties of the analytical solutions. We present some preliminary theoretical results and provide several numerical tests.

26-4 26: Mini-Symposium: Numerical Methods for Parabolic Equations

ON ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR PARABOLIC PROBLEMS Emmanuil H. Georgoulis

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom [email protected]

Spatial discretisation via discontinuous Galerkin (dG) methods is particularly useful for various classes of parabolic problems, such as time-dependent convection-diffusion- reaction and/or spatially fourth order initial/boundary value problems. The suitability of spatial discretisation via dG methods for the aforementioned classes of PDE problems lies in their ability to accommodate discontinuous numerical fluxes across element faces: for the former the ability to introduce (discontinuous) upwind fluxes is crucial, whereas for the latter the lack of H2-conformity is practically desirable. In this talk, some recent experience on the use of adaptive algorithms based on a posteriori error estimation for fully discrete dG methods of interior penalty type for parabolic convection-diffusion and fourth order parabolic problems will be reviewed. The non-conformity of the spatial discretisation introduces some new challenges both in terms of a posteriori error control and in the design of adaptive algorithms, which will be discussed. The talk is based on joint work with Andrea Cangiani, Stephen Metcalfe, and Juha Virtanen from the University of Leicester, UK.

THE HP -ADAPTIVE GALERKIN TIME STEPPING METHOD FOR NONLINEAR DIFFERENTIAL EQUATIONS WITH FINITE TIME BLOW UP B¨arbel Janssen and Thomas P. Wihler

Mathematisches Institut, Universit¨atBern, Sidlerstrasse 5, CH-3012 Bern, Switzerland [email protected]

We consider hp-adaptive Galerkin time stepping methods for nonlinear ordinary dif- ferential equations. The occuring nonlinearity is assumed to be bounded by a constant times the solution to a power β which is larger than one. We prove dual based a posteriori error estimates. Existence of discrete solutions is shown using reconstruc- tion techniques. By means of numerical examples we show that the blow up-is well preserved.

26-5 26: Mini-Symposium: Numerical Methods for Parabolic Equations

MAXIMUM-NORM STRONG APPROXIMATION RATES FOR NOISY REACTION-DIFFUSION EQUATIONS Omar Lakkisa, G.T. Kossioris and M. Romito

Department of Mathematics, University of Sussex, Falmer near Brighton, England UK [email protected]

Pointwise error and maximum norm estimates are important in estimating the risk of rare events happening. I will present new convergence results for the approximation via finite element in space-time and Monte Carlo (or accelerated versions thereof) in the probability to the exact solution process of the stochastic Allen-Cahn equation’s. Con- vergence rates are established for the expected maximum norm in space-time, and are ”strong” in this sense. This improves previous results by Katsoulakis et al (2007) and, although we focus on specific case, our results can be applied to more general reaction– diffusion equations. The results are based on joint work with Georgios Kossioris and Marco Romito.

26-6 26: Mini-Symposium: Numerical Methods for Parabolic Equations

A NEW APPROACH TO ERROR ANALYSIS OF FULLY DISCRETE FINITE ELEMENT METHODS FOR NONLINEAR PARABOLIC EQUATIONS Buyang Li1 and Weiwei Sun2

1Department of Mathematics, Nanjing University, Nanjing, China. [email protected] 2Department of Mathematics, City University of Hong Kong, Hong Kong. [email protected]

Error analysis of fully discrete finite element methods for nonlinear parabolic equa- tions often requires certain time-step size conditions (or stability conditions) such as ∆t = O(hk), which are widely used in the analysis of the nonlinear PDEs from mathe- matical physics, such as the Navier–Stokes equations, viscoelastic flow, the thermistor problem, the time-dependent Ginzburg–Landau equations. Such stability conditions are necessary for explicit schemes, however, due to the various nonlinearities and the coupling of the equations, it is not known whether these time-step size restrictions are necessary for linearized semi-implicit schemes or implicit schemes. In numerical anal- ysis people often assume such stability conditions to derive optimal error estimates, however, whether these conditions are necessary is seldom answered. In this talk, we address this question by exploring a new approach to analyse the error of fully discrete finite element methods for nonlinear parabolic equations. By this new approach, we found that the time-step size restrictions are not necessary for most problems. To illustrate our idea, we analyse two models — the thermistor problem and incompress- ible miscible flow in porous media, with linearized backward Euler scheme for the time discretization. Previous works on the two models all required certain time-step size con- ditions. Our idea is to introduce a system of elliptic PDEs (the time-discrete parabolic n n n n n n PDEs) and split the error into two parts: ku − Uh k ≤ ku − U k + kU − Uh k, where n n n u is the exact solution, Uh is the fully discrete finite element solution, and U is the n solution of the elliptic PDEs. By our choice, the fully discrete solution Uh is also the finite element solution of the elliptic PDEs, i.e. finite element approximations of U n. Therefore, the first part of the error is due to time discretization and depends only on ∆t, the second part of the error is due to the finite element discretization of the elliptic PDEs and depends only on h. With such an error splitting, the time-discretization and the spatial discretization are completely decoupled. Analysis of the second part of the error relies on rigorous analysis of the regularity of U n.

26-7 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine 27 Mini-Symposium: Numerical Methods for Reaction- Transport Equations with Applications in Medicine

Organisers: Jennifer Ryan and Fred Ver- molen

27-1 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine

FINITE ELEMENT ANALYSIS OF THE MECHANO-CHEMICAL REGULATION OF WOUND CONTRACTION IN SURGICAL WOUNDS Etelvina Javierre1, Clara Valero2, Maria Jose Gomez-Benito2 and Jose Manuel Garcia-Aznar2

1 Centro Universitario de la Defensa de Zaragoza, Zaragoza, Spain [email protected] 2 Multiscale in Mechanical and Biological Engineering (M2BE) Aragon Institute of Engineering Research (I3A), Universidad de Zaragoza, Spain

Wound contraction is a highly orchestrated process in which biological and mechanical signals regulate collective cell migration and extracellular matrix synthesis to restore the integrity of a damaged tissue. The aim of this work is to elucidate the effect of wound depth on the contraction kinetics. Hence, only the primary elements in wound contraction are considered (i.e. fibroblasts as the main cell species in the dermis, myofi- broblasts as the contractile phe- notype of fibroblasts, a generic inflammatory growth factor signaling and enhancing cell function, and collagen as the main component of the extracellular matrix). Fur- thermore, we consider a direct coupling between cell function and ECM deformation, including different cell mechanosensing and mechan- otransduction mechanisms. The governing equations are obtained from conservation laws for the cellular and chemical species and the ECM momentum. These conservation laws give rise to a set of non- linearly coupled convection-diffusion-reaction equations that are solved applying the finite element method. Wound morphology and boundary conditions are discussed as well. We limit our analysis to deep and elongated wounds, for which plane strain hypotheses can be as- sumed reducing the model to two spatial dimensions. Thus, the upper part of the domain (representing the interface between the tissue and the external environment) behaves as a free boundary with no displace- ment constrains. The results of this model are compared with earlier works that neglect wound depth (treating the wound as a planar surface). Both approaches give a similar overall con- traction kinetics regarding the contracted area vs time plots. However, taking into account wound depth shows significant changes on the migration pattern and the con- traction evolution.

Acknowledgements: The authors gratefully acknowledge the support of the Spanish Ministry of Economy and Competitiveness through the project DPI2012-32880.

27-2 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine

PRESENTATION OF RESULTS OF FINITE-ELEMENT ANALYSES ON A TWO-DIMENSIONAL MECHANOCHEMICAL MODEL FOR DERMAL WOUND HEALING D.C. Koppenola and Fred J. Vermolenb

DIAM, Delft University of Technology, Delft, The Netherlands [email protected], [email protected]

The healing of full-thickness dermal wounds involves a complicated sequence of spa- tially and temporally coordinated processes that can be classified roughly into four consecutive, partly overlapping phases: haemostasis, inflammation, proliferation, and remodeling. One of the key processes that takes place during the proliferative phase is the contraction of the wound. During this process the wound boundaries are drawn inwards by biomechanical mechanisms so that the size of the injury is reduced. This phenomenon is an important and intrinsic feature in the healing process and it is usually beneficial when it is well-balanced. If this balance is disrupted however, then this may cause delayed and / or impaired healing in case of insufficient contraction, while exces- sive contraction may induce low quality repair with substantial scarring. Even though intensive research over the last few decades has produced much knowledge about the biomechanical mechanisms underlying wound contraction and its associated patholo- gies, there is much that remains understood incompletely about these mechanisms and the etiology of the pathologies that may develop. In order to gain more insight into the mechanisms underlying wound contraction, a new deterministic model, consisting of a system of nonlinearly coupled parabolic-hyperbolic partial differential equations, has been developed recently by Murphy et al. [1]. This model allows a detailed evaluation of the effects of strong interactions between me- chanical changes and some of the most important biological entities involved in the contraction of full-thickness dermal wounds. This evaluation is accomplished by de- scribing the interactions between (myo-)fibroblasts, two growth factors, collagen, the enzyme collagenase and a mechanical force balance as a result of the physico-chemical properties of the extracellular matrix and the cell-generated traction forces. Murphy et al. assume that the wound is long, thin and much longer than it is deep and therefore it is appropriate for them to consider a one-dimensional representation. After obtaining some results from a numerical analysis of this one-dimensional model, they made a comparison between the model predictions and experimental data on human dermal wound healing which shows that all the essential features are well matched. Given this nice match between the one-dimensional model predictions and the experi- mental data, and the substantial impact the geometry of the wound has on the healing process, we investigate the influence of geometry. To this end, we extended the model of Murphy et al. to two spatial dimensions and we used various initial conditions to simu- late several wound geometries. We will present some of the results from finite-element analyses.

27-3 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine

References

[1] Murphy, K.E., Hall, C.L., Maini, P.K., McCue, S.W., McElwain, D.L.S., A Fi- brocontractive Mechanochemical Model of Dermal Wound Closure Incorporating Realistic Growth Factor Kinetics. Bulletin of Mathematical Biology 74, pp. 1143- 1170, 2012.

MATHEMATICAL MODELLING AND NUMERICAL SIMULATIONS OF ACTIN DYNAMICS IN THE EUKARYOTIC CELL Anotida Madzvamuse1a, Uduak George1 and Angelique St´ephanou2

1University of Sussex, School of Mathematical and Physical Sciences, Pevensey III, 5C15. BN1 9QH. Brighton, UK. [email protected] 2UJF-Grenoble 1, CNRS, Laboratoire TIMC-IMAG UMR 5525, DyCTiM research team, Grenoble, F-38041, France

In this talk I will present a model for cell deformation and cell movement that couples the mechanical and biochemical properties of the cortical network of actin filaments with its concentration. Actin is a polymer that can exist either in filamentous form (F- actin) or in monometric form (G-actin) (Chen et al., in Trends Biochem Sci 25:1923, 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrishnan et al., in Recent Res Devel Biophys 5:3969, 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskele- ton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its biochemical dynamics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al., in J Comput Phys 190:478500, 2003). Furthermore, by assuming slow deformations of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the mathematical model is able to describe the complex cell deformations typically observed in exper- imental results. Our numerical results illustrate cell expansion, cell contraction, cell translation and cell relocation as well as cell protrusions in agreement with experimen- tal observations. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter. Cell migration plays a critical and pivotal role in a variety of biological and biomedical disease processes and is important for emerging areas of biotechnology which focus on cellular transplantation and the manufacture of artificial tissues and surfaces, as well as for the development of new therapeutic strategies for controlling invasive tumor cells.

27-4 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine

ANALYZING THE TREATMENT OF A BACTERIAL INFECTION IN A WOUND USING OXYGEN THERAPY Richard Schugart

Department of Mathematics, West Kentucky University, USA [email protected]

In this work, a reduced mathematical model is to be presented for the treatment of a bacterial infection in a wound using oxygen therapy. The model considers the interactions of neutrophils and bacteria in a wound combined with the use of hyperbaric and topical oxygen therapies to aide in the removal of bacteria from the wound. Both a traveling wave solution and numerical simulations of the model will be presented. The application of optimal control theory for the model will be discussed.

27-5 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine

A SEMI–STOCHASTIC MODEL FOR THE IMMUNE RESPONSE SYSTEM Fred J. Vermolen

Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands [email protected]

The immune response system is a vital defense mechanism for human beings. Defects can be caused by a shortage of white blood cells (such as lymphocytes, monocytes, neutrophils, or macrophages), or by an impairment of the transmittivity of the venule (small blood vessel) wall. In case of a (bacterial) infection, bacterial secretion alerts macrophages that secrete chemokines that are detected by the endothelial cells that constitute the venule walls. From there, cytokines are released and detected by the white blood cells that are advected in the blood stream. The white blood cells move towards the venule walls and leave the venule by transmigration through the venule wall. In this presentation, we will discuss a semi-stochastic model for the immune re- sponse system. The model is based on simple principles, such as Poisseuille flow through venules, Brownian motion of bacteria, fundamental solutions for bacterially secreted, and migration and deformation of white blood cells. The gradient of a chemical stim- ulus acts as the driving force of white blood cell migration and deformation. This gradient is obtained by a superposition of point sources that mimic bacterial secre- tion. The white blood cells are modeled to be elastic and hence spring forces define the driving force of the white blood cell to get reshaped in its original geometry. The cellular migration and deformation is modeled through straightforward formalisms for curve shape evolution, and not by the solution of partial differential equations. Hence the model is based on systems of ordinary (stochastic) differential equations and finite element strategies are not used. We show the mathematical formulation of the model, as well as some implications of the model, such as a parameter variation based on statistical principles.

27-6 27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications in Medicine

MULTISCALE MODELS OF TUMOR CELLS: FROM IN-VITRO AGGREGATES TO IN-VIVO VASCULARIZED TUMORS Irene Vignon-Clementela, Nick Jagiella and Dirk Drasdo

INRIA, Paris Rocquencourt, France a [email protected]

This work aims at better understanding the dynamic interplay between tumor cells and their environment, and its assessment by medical imaging, to in fine improve treatment. A multiscale model was built that represents the generic features of such an interplay. Tumor cells are modeled individually, interacting and evolving by rules that depend on their environment. In turn, the cells modify their environment (nutrients, etc), which is represented by nonlinear continuum reaction diffusion equations. Additional bricks were added to model the vascularization that provides nutrients to the tumor cells when studying the in-vivo situation. Angiogenesis & vessel remodeling induced by the over proliferating and hypoxic tumor cells, have been included in this generic model to study this interplay via the molecular scale. A second part of the work has been to get closer to real settings. To study the in-vitro multicellular growth of a specific lung cancer cell type, the model has been refined based on experimental data. Selected mechanisms have been proposed to explain the spatio-temporal information of these experimental data. Finally, we will present some challenges and novel research axes to go to the in-vivo scale. Numerical methods and results will be presented throughout the talk.

27-7 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

28 Mini-Symposium: Numerical methods for con- tact and other geometrically non-linear prob- lems

Organisers: Alexey Chernov and Matthias Maischak

28-1 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

PARALLEL SOLUTION OF CONTACT SHAPE OPTIMIZATION PROBLEMS WITH COULOMB FRICTION BASED ON DOMAIN DECOMPOSITION P. Beremlijskia Tom´aˇsBrzobohat´yb Tom´aˇsKozubekc and Alexandros Markopoulosd

VSB-Technicalˇ University of Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic. [email protected], [email protected], [email protected], [email protected]

We shall first briefly introduce the FETI based domain decomposition methodology adapted to the solution of multibody contact problems with Coulomb friction. These problems play a role of the state problem in contact shape optimization problems with Coulomb friction. We use a modification of FETI that we call Total FETI, which imposes not only the compatibility of a solution across the subdomain interfaces, but also the prescribed displacements. For solving a state problem we use the method of successive approximations. Each iterative step of the method requires us to solve the contact problem with given friction. As a result, we obtain a convex quadratic programming problem with a convex separable nonlinear inequality and linear equal- ity constraints. For the solution of such problems we use a combination of inexact augmented Lagrangians in combination with active set based algorithms. The discretized problem with Coulomb friction has a unique solution for small coef- ficients of friction. The uniqueness of the equilibria for fixed controls enables us to apply the so-called implicit programming approach. Its main idea consists in minimization of a nonsmooth composite function generated by the objective and the control-state mapping. The implicit programming approach combined with the differential calculus of Clarke was used for a discretized problem of 2D shape optimization. There is no possibility to extend the same approach to the 3D case. To get subgradient information needed in the used numerical method we use the differential calculus of Mordukhovich. Application of Total FETI method to the solution of the state problem and sensitivity analysis allows massively parallel solution of these problems. The effectiveness of our approach is demonstrated by numerical experiments.

References

[1] P. Beremlijski, J. Haslinger, M. Koˇcvara, R. Kuˇceraand J. Outrata: Shape Op- timization in Three-Dimensional Contact Problems with Coulomb Friction. In: SIAM Journal on Optimization 20/1, 2009, pp. 416-444.

[2] P. Beremlijski, T. Brzobohat´y,T. Kozubek, A. Markopoulos and J. Outrata: Par- allel solution of contact shape optimization problems with Coulomb friction based on domain decomposition. In: WIT Transactions on the Built Environment 124, 2012, pp. 285-295.

28-2 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

PARALLEL SOLUTION OF ELASTO-PLASTIC PROBLEMS Martin Cerm´akˇ a and Michal Mertab

Centre of Excellence IT4Innovations, VSB-TUˇ Ostrava, Czech Republic. [email protected], [email protected]

In this work we present the parallel solution of elasto-plastic problems. We assume the von Mises plastic criterion with kinematic hardening and the associated plastic flow rule. For the time discretization we use the implicit Euler method and the corre- sponding one-time-step problem is formulated with respect to unknown displacement. For the space discretization we use the finite element method and we parallelize the resulting problem using the Total-FETI method. Our parallel implementations are based on the Trilinos and the PETSc software frameworks. Their performance and scalability are compared on 2D and 3D bench- marks. The scalability tests were carried out using the HECToR supercomputer at EPCC, UK.

28-3 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

CONVERGENCE ANALYSIS FOR MULTILEVEL VARIANCE ESTIMATORS IN MULTILEVEL MONTE CARLO METHODS AND APPLICATION FOR RANDOM OBSTACLE PROBLEMS Alexey Chernova and Claudio Bierigb

Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Germany. [email protected], [email protected]

The Multilevel Monte Carlo Method (MLMC) is a recently established sampling ap- proach for uncertainty propagation for problems with random parameters. Under cer- tain assumptions, MLMC allows to estimate e.g. the mean solution and k-point corre- lation functions at essentially the same overall computational cost as the cost required for solution of one forward problem for a fixed deterministic set of parameters. However, in many practical applications estimation of the variance (along with the mean) is the main goal of the computations. In this case the variance can be potentially computed from correlation functions in the post-processing step. This approach has two drawbacks:

1. Optimal complexity approximation of correlation functions involves quite cum- bersome sparse tensor product constructions. It is desirable to avoid it if the variance is the aim of the computation.

2. Computation of the variance from the 2-point correlation function is prone to numerical instability, specially in the case of small variances.

Much less is known about direct estimation of the variance, potentially overcoming these difficulties. In this talk we present new convergence theorems for the multilevel variance estimators. As a result, we prove that under certain assumptions on the parameters, the variance can be estimated at essentially the same cost as the mean, and consequently as the cost required for solution of one forward problem for a fixed deterministic set of parameters. We comment on fast and stable evaluation of the estimators suitable for parallel large scale computations. The suggested approach is applied to a class of scalar random obstacle problems, a prototype of contact between deformable bodies. In particular, we are interested in rough random obstacles modelling contact between car tires and variable road surfaces. Numerical experiments support and complete the theoretical analysis.

28-4 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

SCALABLE ALGORITHMS AND CONDITIONING OF CONSTRAINTS ARISING FROM VARIATIONALLY CONSISTENT DISCRETIZATION OF CONTACT PROBLEMS ZdenˇekDost´al, Tom´aˇsKozubek and Oldˇrich Vlach

VSB–Technicalˇ University of Ostrava, CZ-70833 Ostrava, Czech Republic [email protected]

The results related to the development of theoretically supported scalable algorithms for the solution of large scale transient contact problems of elasticity will be briefly reviewed [1]. The algorithms that were originally developed for matching discretiza- tion combine the Total FETI/BETI based domain decomposition methods adapted to the solution of 2D and 3D multibody contact problems of elasticity with optional preconditioning by conjugate projector or dual scaling with our in a sense optimal al- gorithms for the solution of resulting quadratic programming or QPQC problems. The theoretical results are qualitatively the same as the classical results on scalability of FETI/BETI for linear elliptic problems, i.e., the inequality constraints are treated in a sense for free. In this presentation we discuss generalization of these results to the problems discretized by non-matching grids. We consider implementation of the non- penetration condition by the variationally consistent discretization introduced recently by B. I. Wohlmuth [2] and study performance of related algorithms. We give bounds on the spectrum of the related matrices for some mortar discretizations and compare them with the numerical values obtained for some special cases [3]. We also provide results of numerical experiments showing that variationally consistent discretization preserve efficiency of the scalable TFETI/TBETI solvers.

References

[1] Z. Dost´al,T. Kozubek, T. Brzobohat´y,A. Markopoulos, and O. Vlach, Scalable TFETI with optional preconditioning by conjugate projector for transient contact problems of elasticity, Computer Methods in Applied Mechanics and Engineering 247–248 (2012) 37–50.

[2] B. I. Wohlmuth Variationally consistent discretization schemes and numerical algorithms for contact problems, Acta Numerica (2012) 569–734.

[3] Z. Dost´al,T. Kozubek, O. Vlach, and T. Brzobohat´y, On scalable algorithms and conditioning of constraints arising from variationally consistent discretization of contact problems.

28-5 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

PARALLEL SOLUTION OF CONTACT PROBLEMS BASED ON TFETI ZdenˇekDost´ala, Tom´aˇsBrzobohat´yb, Tom´aˇsKozubekc, Alexandros Markopoulosd and Oldˇrich Vlache

VSB-Technicalˇ University of Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic. [email protected], [email protected], [email protected], [email protected], [email protected]

In our contribution we briefly review the TFETI based domain decomposition method- ology adapted to the solution of 2D and 3D multibody contact problems. For the solution of the resulting quadratic programming problems our in a sense optimal al- gorithms are used and we will present them together with their powerful ingredients. Our results obtained for elastic contact problems are extended to the contact problems with non-matching grids which necessarily arise, e.g., in the solution of transient or shape optimization problems. We consider both standard engineering approaches such as node to segment, or mortar elements. The aim is to get the constraint matrix B with nearly orthogonal rows, which is required assumption of our algorithms. The simple bounds on the singular values of the resulting matrix B as well as the results of numeri- cal experiments, including both the academic examples and some problems of practical interest will be presented. We conclude that the normalized orthogonal mortars pro- posed by Wohlmuth can be used to approximate the non-penetration conditions in a way that complies with the requirements of the FETI methods.

References

[1] Z. Dost´al,D. Hor´ak,R. Kuˇcera: Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE, Communications in Numerical Methods in Engineering 22 (2006) 1155–1162.

[2] Z. Dost´al:Optimal quadratic programming algorithms: with applications to vari- ational inequalities, Springer, New York, 2009.

[3] Z. Dost´al,T. Kozubek, A. Markopoulos, M. Menˇs´ık: Cholesky decdomposition and a generalized inverse of the stiffness matrix of a floating structure with known null space, Applied Mathematics and Computation 217 (2011) 6067-6077.

[4] R. Kuˇcera,T. Kozubek, A. Markopoulos: On large-scale generalized inverses in solving two-by-two block linear systems, submitted to Linear Algebra and Its Ap- plications (2012).

[5] T. Kozubek, Z. Dost´al, T. Brzobohat´y, O. Vlach, A. Markopoulos: Scalable TFETI with optional preconditioning by conjugate projector for transient Com- puter, Methods in Applied Mechanics Engineering, 2012, 247-248, 37-50

28-6 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

[6] B. I. Wohlmuth: Variationally consistent discretization schemes and numer- ical algorithms for contact problems. Acta Numerica, 20 (2011), 569734.

28-7 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

LOCAL AVERAGING OF CONTACT WITH NON MATCHING MESHES Guillaume Drouet1 and Patrick Hild2

1 LaMSID - Laboratoire de M´ecaniquedes Structures Industrielles Durables, UMR 8193, EDF, CNRS, CEA, France, [email protected] 2 Institut de Math´ematiquesde Toulouse, UMR 5219, Universit´eToulouse 3, CNRS, France, [email protected]

Finite element methods are currently used to approximate the unilateral contact prob- lem. Such a problem shows a nonlinear boundary condition, which roughly speaking requires that (a component of) the solution u is nonpositive (or equivalently nonnega- tive) on (a part of) the boundary of the domain Ω. This nonlinearity leads to a weak formulation written as a variational inequality which admits a unique solution) and the regularity of the solution shows limitations whatever the regularity of the data is. A consequence is that only finite element methods of order one and of order two are of interest. In this talk we limit ourselves to the finite element method of order one in two and three space dimensions and we consider a discrete contact condition which requires, in the case of a sole body in contact with a rigid foundation, that the approximate solution uh is nonpositive in average on some local patches comprising several contact elements that form a partition of the contact zone. The corresponding discrete convex cone of admissible functions is given by  Z  Kh = vh ∈ V h : vh dΓ ≤ 0 ∀T m ∈ T M , N T m where T M is a macro-mesh. The discrete convex cone of admissible solutions is not a subset of the continuous convex cone of admissible solutions. So, in order to achieve the error analysis, we have to bound two terms, the approximation error and the consistency error. First, we show that if the macro-mesh T m satisfies a reasonnable technical assump- tion, we are able to construct an average preserving operator wich leads us to an optimal approximation error so that the convergence error of the method will be given by the consistency error. In dimension 2, this choice gives us slightly better theoretical results than the existing ones.

Then, we show that this approach can be easily extended to multi body contact with non matching meshes in two and three space dimensions which is the goal of this method. Finally, we show that the assumption made on T M could be linked to the inf-sup conditions when considering the mixed method associated to the variational inequality.

28-8 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

BE/FE APPROXIMATION OF HIGHER ORDER FOR NONSMOOTH PROBLEMS, EFFECTIVE QUADRATURE, AND TIME DISCRETIZATION BY IMPLICIT RUNGE-KUTTA METHODS

Joachim Gwinner

Institute of Mathematics, Department of Aerospace Engineering, University of the Federal Army Munich, D - 85577 Neubiberg, Germany [email protected]

In this talk we address higher order approximation in the FEM , BEM or in FEM-BEM coupling for unilateral problems including nonsmooth friction-type functionals. Such an approximation leads to a nonconforming discretization scheme.To fix ideas consider the friction-type functional Z j(v) = g|v| ds .

ΓC Then in contrast to previous related work we approximate such a functional using Gauss-Lobatto quadrature by its e.g. p-approximation

p X X p+1 p+1 jp(v) = ωj v ◦ Fe(ξj ) e∈Sh,e⊂ΓC j=0 and take the quadrature error of the friction functional into account of the error anal- ysis. Moreover we are concerned with full space time discretization of related nons- mooth parabolic and evolutionary inequality problems employing implicit Runge-Kutta metods for time discretization. This talk is based on the recent work [1, 2, 3].

References

[1] J. Gwinner, On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction, Appl. Numer. Math. 60 (2010) 689 – 704.

[2] J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca fric- tion in plane linear elastostatics, J. Comput. Appl. Math. (2013, in press).

[3] J. Gwinner & M. Thalhammer, Full discretisations for nonlinear evolutionary in- equalities based on stiffly accurate Runge-Kutta and hp-finite element methods, to appear.

28-9 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

A DISCRETIZATION FOR DYNAMIC LARGE DEFORMATION CONTACT PROBLEMS OF NONLINEAR HYPERELASTIC CONTINUA Ralf Kornhuber1a, Oliver Sander2 and Jonathan Youett1c

1Department of Mathematics and Computer Science, Freie Universit¨atBerlin, Germany. [email protected], [email protected], 2Department of Mathematics, IPGM, RWTH Aachen, Germany. [email protected]

In this talk we present a discretization for dynamic large deformation frictionless con- tact problems of nonlinear hyperelastic continua. The equations of motion are derived by a nonsmooth Hamilton principle which results in a differential inclusion in the framework of generalized gradients [F.H. Clarke, Optimization and Nonsmooth Anal- ysis, 1983]. We use a mortar method with dual basis functions for the discretization of the contact constraints which are known to be very robust. For the time discretiza- tion of the inclusion we apply a contact-stabilized midpoint rule which leads to spatial problems that can be reformulated as minimization problems. These non-convex min- imization problems are then solved using a quasi Newton SQP method. The efficiency of SQP methods mainly depends on the quality of the solver for the quadratic sub- problems. In our proposed method we use an approximation of the linearized contact constraints for the subproblems, which allows us to apply a special basis transformation that leads to a decoupling of the constraints [B. Wohlmuth and R. Krause, Monotone Methods on Non-Matching Grids for Nonlinear Contact Problems,2003]. The result- ing quadratic problem is then solved fast and efficiently using a monotone multigrid method. Numerical results illustrate the energy and convergence behaviour of the proposed scheme.

28-10 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

PARALLEL LEVEL SET METHODS FOR LARGE DEFORMATION CONTACT PROBLEMS Rolf Krause1a, Valentina Poletti1b, Roberto Croce1c and Petr Kotas2

1Institute of Computational Science, University of Lugano, Switzerland [email protected], [email protected] [email protected] 2Department of Applied Mathematics, Technical University of Ostrava, Czech Republic

In this talk, we present and discuss a new approach for the efficient and highly parallel computation of the distance between a deformable body and an obstacle using level-set methods. The numerical solution of contact problems involving large deformations requires an acurate treatment of the non-linear contact constraints. Since these constraints depend on the current configuration, during any iterative solution process they have to be updated repeatedly. Unfortunately, straight-forward approaches as, e.g. the direct computation of the closest point projection, are computationally intensive and possibly error-prone. In view of the potential ambiguity of closest points we therefore follow an alternative approach where the contact condition is described by a signed distance function to the obstacles boundary. This approach allows for a fast and robust computation of the distance to the obstacle for different configurations by simply evaluating the precomputed signed-distance function. However, in view of the ever increasing parallelism on modern computers and super- computers, the signed-distance function should be computable in parallel. In this talk, we first discuss the difficulties when aiming at a parallel computation of the signed- distance function. We then present our new and fully parallel fast marching method, which allows for the efficient and parallel computation of the signed-distance function. Numerical examples illustrating the performance of our approach will be given.

28-11 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

ERROR ESTIMATORS FOR A PARTIALLY CLAMPED PLATE PROBLEM WITH BOUNDARY ELEMENTS Matthias Maischak

BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge, UK. [email protected]

The biharmonic equation models the bending of a thin elastic plate. Restricting the corresponding minimization problems on a convex subset of possible boundary condi- tions, cf. [2], describing restrictions on the clamping of the plate boundary, we obtain a variational inequality. Using the fundamental solution we obtain a symmetric inte- gral operator representation [5]. The higher regularity requirements of the biharmonic operator lead to the usage of C1 basis functions, as well as to a C1 regular representa- tion of the boundary. We will first present a high-order numerical quadrature scheme, suitable for a hp-method on curved boundaries and second, we will derive a-posteriori error estimates based on a hierarchical decomposition, cf. [8, 9]. Several numerical examples underline the theoretical results.

References

[1] L. Caffarelli, The obstacle problem, (1998).

[2] L. A. Caffarelli, A. Friedman, and A. Torelli, The two-obstacle problem for the biharmonic operator, Pacific Journal of Mathematics, 103 (1982), pp. 325– 335.

[3] F. Cuccu, B. Emamizadeh, and G. Porru, Optimization problems for an elastic plate, Journal of Mathematical Physics, 47 (2006).

[4] W. Han, D. Hua, and L. Wang, Nonconfoming finite element methods for a clamped plate with elastic unlateral obstacle, Journal of Integral Equations and Applications, 18 (2006), pp. 267–284.

[5] G. Hsiao and W. L. Wendland, Boundary Integral Equations, vol. 164 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 2008.

[6] Y. Jeon and W. McLean, A new boundary element method for the biharmonic equations with dirichlet boundary conditions, Advances in Computational Mathe- matics, 19 (2003), pp. 339–354.

[7] M. Maischak, P. Mund, and E. P. Stephan, Adaptive multilevel bem for acoustic scattering, Comput.Methods Appl.Mech.Engrg., 150 (1997), pp. 351–367.

[8] M. Maischak and E. P. Stephan, Adaptive hp versions of bem for Signorini problems, Applied Numerical Methods, 54 (2005), pp. 425–449.

[9] , Adaptive hp-versions of boundary element methods for elastic contact prob- lems, Comput. Mech., 39 (2007), pp. 597–607.

28-12 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

[10] C. Tosone and A. Maceri, The clamped plate with elastic unilateral obstacles: a finite element approach, Math. Models Methods Appl. Sci., 13 (2003), pp. 1231– 1243.

OPTIMAL ACTIVE-SET AND SPECTRAL ALGORITHMS FOR THE SOLUTION OF 3D CONTACT PROBLEMS WITH ANISOTROPIC FRICTION Luk´aˇsPosp´ıˇsila, ZdenˇekDost´alb and Tom´aˇsKozubekc

FEI VSB-Technicalˇ University Ostrava, Tˇr17 listopadu, CZ-70833 Ostrava, Czech Republic [email protected], [email protected], [email protected]

The formulation of a contact problem with anisotropic friction in terms of contact stresses leads to the minimization of a strictly convex quadratic function subject to separable inequality ellipsoidal constraints and non-penetration inequality linear con- straints. For solving this optimizing problem, we present a modification of our recently developed in a sense optimal MPGP algorithms [1]. These active-set based algorithms explore the faces by the conjugate gradients and change the active sets and active variables by the gradient projection with the constant steplength, which guarantees the monotone descend of the cost function. We present also a modification, which replaces the con- stant steplength by the projected Barzilai-Borwein (spectral) method [2], that rapidly decreases the number of projection steps.

The efficiency of our algorithms is illustrated on the solution of a 3D contact problem of one cantilever beam and rigid obstacle in mutual contact with the Tresca friction.

z

Γ F Fg F ΓD

x ΓC rigid obstacle y

For solving such a large-scaled problem, we use a Total FETI (TFETI) based domain decomposition, which involves additional equality linear constraints. Here MPGP is used in the inner loop of the Semi-monotonic augmented Lagrangian method.

28-13 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

References

[1] Dost´al,Z., Kozubek, T.: An optimal algorithm and superrelaxation for minimiza- tion of a quadratic function subject to separable convex constraints with applications, Mathematical Programming, vol. 135, pp. 195–220, 2012.

[2] Birgin, E.G., Mart´ınez, J.M., and Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization 10, pp. 1196– 1211, 2000.

HP -ADAPTIVE FEM WITH BIORTHOGONAL BASIS FUNCTIONS FOR ELLIPTIC OBSTACLE PROBLEMS Andreas Schr¨oder1 and Lothar Banz2

1 Department of Mathematics, University of Salzburg, Austria, [email protected] 2 Institute of Applied Mathematics, Leibniz University Hanover, Germany, [email protected]

In this talk, the discretization of a non-symmetric elliptic obstacle problem with hp- adaptive conforming finite elements is discussed. For this purpose, a higher-order mixed finite element discretiziation is introduced where the dual space is discretized via biorthogonal basis functions. hp-adaptivity is realized via automatic adaptive mesh re- finement based on a posteriori error estimates. The use of biorthogonal basis functions leads to an algebraic system including box constraints and componentwise complemen- tarity conditions. This structure is exploited to apply efficient semismooth Newton methods using a penalized Fischer-Burmeister NCP-function in each component. To include meshes with hanging nodes and varying polynomial degrees resulting from hp- adaptive mesh refinements the use of appropriate connectivity matrices is proposed. Several numerical experiments confirm the applicability of the hp-adaptive scheme.

28-14 28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

HIGH ORDER BEM FOR FRICTIONAL CONTACT PROBLEMS Ernst P. Stephan

Institute for Applied Mathematics, Leibniz Universit¨atHannover, Hannover, Germany. [email protected]

We consider frictional contact for non-linear elasticity in R2. We use the Poincar´e- Steklov operator, which realizes the Dirichlet-to-Neumann map, and represent the negative of the unknown normal traction on the contact boundary by a Lagrange mul- tiplier. Herewith we derive a mixed formulation which is equivalent to a variational inequality on the contact boundary, where the non-penetration condition is incorpo- rated in the convex set of admissible ansatz and test functions. Both formulations are uniquely solvable. We use Gauss-Lobatto-Lagrange basis functions on a regular mesh on the contact boundary for the primal variable and biorthogonal basis functions of the same degree on the same mesh for the Lagrange multiplier. We present a reliable and efficient a posteriori error estimate of residual type for the Galerkin solution of the mixed formulation. The discrete mixed system is solved by the semi-smooth New- ton algorithm in combination with a penalized Fischer-Burmeister complementarity function taking care of the contact condition. Numerical experiments are given which support our theoretical results.

28-15 29: Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

29 Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

Organisers: Susanne Brenner, Klaus B¨ohmer and Michael Neilan

29-1 29: Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

PSEUDO TRANSIENT CONTINUATION AND TIME MARCHING METHODS FOR MONGE-AMPERE` TYPE EQUATIONS Gerard Awanou

Department of Mathematics, Statistics, and Computer Science University of Illinois, Chicago 322 Science and Engineering Offices (SEO) M/C 249, 851 South Morgan Street, Chicago, IL 60607-7045, USA [email protected]

We present two numerical methods for the fully nonlinear elliptic Monge- Amp`ere equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods perform well across a wide range of dis- cretizations: C1 conforming approximations, standard Lagrange finite element method, standard finite difference method, etc. We will present recent results on the conver- gence of the iterative methods for solving the nonlinear system of equations resulting from the discretizations and prove the convergence of the discretizations for smooth solutions. We give numerical evidence that the methods are also able to capture the viscosity solution of the Monge-Amp`ereequation. Even in the case of the degener- ate Monge-Amp`ereequation, the time marching method appears also to compute the viscosity solution.

GENERAL FULL DISCRETIZATIONS FOR CENTER MANIFOLDS, HERE FOR FULLY NONLINEAR EQUATIONS AND FEMS Klaus B¨ohmer

Fachbereich Mathematik und Informatik, Philipps University, Marburg, Germany [email protected]

Dynamical systems are often studied for parabolic PDEs and their discretization. The nonlinear elliptic parts are either equations or system of order 2 or 2m, m > 1. Space and time discretization methods, so called full discretizations, are necessary to deter- mine the dynamics on stable and center manifolds for these problems. My general theory essentially starts with the standard space discretization methods used for nonlinear elliptic PDEs. The coefficients of the Taylor expansion of a space discretized center manifold and its normal form converge to those of the original center manifold. Then standard, e.g., Runge–Kutta, or geometric time discretization methods can be applied to the discrete center manifold, a small dimensional system of ordinary differential equations. These results are applicable to any of the recent FEMs for fully nonlinear elliptic/parabolic PDEs, some of them presented in this mini symposium.

29-2 29: Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

NUMERICAL SOLUTION OF MONGE-AMPERE` EQUATION ON DOMAINS BOUNDED BY PIECEWISE CONICS Oleg Davydov1 and Abid Saeed2

1Department of Mathematics and Statistics, University of Strathclyde, Scotland, [email protected], 2Department of Mathematics, Kohat University of Science & Technology, Kohat, Pakistan, [email protected]

We introduce new C1 polynomial finite element spaces for curved domains bounded by piecewise conics using Bernstein-B´eziertechniques. These spaces are employed to solve fully nonlinear elliptic equations. Numerical results for several test problems for the Monge-Amp`ereequation on domains of various smoothness orders endorse theoretical error bounds given previously by K. B¨ohmer.

DISCONTINUOUS GALERKIN FINITE ELEMENT DIFFERENTIAL CALCULUS AND APPLICATIONS Xiaobing Feng

Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, U.S.A. [email protected]

In this talk I shall first present a newly developed discontinuous Galerkin finite ele- ment differential calculus theory for approximating weak (or distributional) derivatives of broken Sobolev functions. After the definition is introduced, various properties and calculus rules (such as product and chain rule, integration by parts formula and di- vergence theorem) for the numerical derivatives will be outlined. I shall then discuss how the proposed discontinuous Galerkin finite element differential calculus can be (conveniently) used to build discretization methods for linear and nonlinear (including fully nonlinear) PDEs. This is a jointly work with Michael Neilan of University of Pittsburgh and Tom Lewis of the University of Tennessee at Knoxville.

29-3 29: Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

A FINITE ELEMENT METHOD FOR HAMILTON-JACOBI-BELLMAN EQUATIONS Max Jensen1 and Iain Smears2

1Department of Mathematics, University of Sussex, England [email protected] 2Mathematical Institute, University of Oxford, England [email protected]

Hamilton-Jacobi-Bellman equations describe how the cost of an optimal control prob- lem changes as problem parameters vary. This talk will address how Galerkin methods can be adapted to solve these equations efficiently. In particular, it is discussed how the convergence argument by Barles and Souganidis for finite difference schemes can be extended to Galerkin finite element methods to ensure convergence to viscosity solutions. A key question in this regard is the formulation of the consistency condition. Due to the Galerkin approach, coercivity properties of the HJB operator may also be satisfied by the numerical scheme. In this case one achieves besides uniform also strong H1 convergence of numerical solutions on unstructured meshes.

ADAPTIVITY AND FULLY NONLINEAR PROBLEMS Omar Lakkis1 and Tristan Pryer2

1 Department of Mathematics, University of Sussex, Brighton, England. [email protected] 2 School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, England. [email protected]

In this talk we discuss the development of residual based h–adaptivity for a class of finite element discretisation of fully nonlinear problems introduced in [LP:2011]. We pay particular attention to the Monge Amp`ereequation, the Infinity Laplacian and Pucci’s equation, looking at the efficiency of the resulting estimator for these problems.

References

[LP:2011] Lakkis, Omar and Pryer, Tristan. A nonvariational finite element method for nonlinear elliptic problems. Submitted - tech report available on ArXiV http://arxiv.org/abs/1103.2970, 2011.

29-4 29: Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

FINITE ELEMENT METHODS FOR THE MONGE-AMPERE` EQUATION Michael Neilan

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA [email protected]

In this talk, I will discuss finite element methods for the fully nonlinear Monge-Amp`ere equation. The main feature of our discretizations is that their linearizations are co- ercive over the finite element space. I will describe a simple procedure to construct such schemes and briefly discuss the difficulties in the convergence analysis. Finally, numerical examples for a series of benchmark problems will be presented.

29-5 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

30 Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

Organisers: Ahmed H. Elsheikh, Ben Ganis and Mary Wheeler

30-1 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

MULTIPHASE FLOW THROUGH POROUS MEDIA: AN ADAPTIVE CONTROL VOLUME FINITE ELEMENT METHOD Peyman Mostaghimia, James R. Percival, Brendan S. Tollit, Stephen J. Neethling, Gerard J. Gorman, Matthew D. Jackson, Christopher C. Pain and Jefferson L.M.A. Gomes

Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK [email protected]

Numerical simulation of multiphase flow in porous media is of importance in a wide range of applications in science and engineering. We present the formulation for an incompressible adaptive control volume finite element method for accurate and efficient modelling of flow in porous media. The saturation equation is spatially discretized us- ing a node centred control volume method on an unstructured finite element mesh. The pressure equation is spatially discretized using a continuous control volume finite ele- ment method (CVFEM) to achieve consistency with the discrete saturation equation. The proposed scheme is CV-wise locally mass conservative while geometric flexibility of the finite element method is retained. The numerical simulation is implemented in the framework of Fluidity, an open source and general purpose fluid flow simulator capable of solving a number of different governing equations on arbitrary unstructured meshes. The scheme is equipped with dynamic anisotropic mesh adaptivity to update the mesh resolution to capture the evolving features of flow through the porous medium. It uses metric advection between adaptive meshes in order to predict the future density of mesh. We demonstrate the advantage of using dynamic mesh adaptivity for flow in porous media. In addition, we compare the obtained results for simulation of fluid flow in some challenging benchmark geometries against conventional finite-difference simu- lations. We also apply the method for large-scale simulation of heap leaching process for mineral recovery applications. The obtained results demonstrate the capability of the scheme for flow simulation in complex geometries with high spatial accuracy at low computational cost through the use of anisotropic mesh adaptivity.

30-2 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

MULTIPOINT FLUX DOMAIN DECOMPOSITION TIME-SPLITTING METHODS ON GENERAL GRIDS Andr´esArrar´as1a, Laura Portero1b and Ivan Yotov2

1Departamento de Ingenier´ıaMatem´aticae Inform´atica, Universidad P´ublicade Navarra, Pamplona, Spain [email protected], [email protected] 2Department of Mathematics, University of Pittsburgh, Pittsburgh, USA [email protected]

In this work, we propose and analyze efficient discretizations for mixed formulations of evolutionary diffusion problems on general grids. The spatial approximation is based on the multipoint flux mixed finite element method, which allows for local flux elimination by using suitable finite element spaces and special quadrature rules. As a result, we obtain a cell-centered pressure system on triangular, quadrilateral, tetrahedral and hexahedral meshes. Such a system is subsequently partitioned via an overlapping domain decomposition splitting technique. A proper combination of this technique with multiterm fractional step diagonally implicit Runge–Kutta methods reduces the global system to a collection of uncoupled subdomain problems that can be solved in parallel. The fully discrete scheme is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. We derive a priori error estimates for both the semidiscrete and fully discrete formulations, and further illustrate the theoretical results with numerical experiments.

30-3 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

AN OPTIMIZATION APPROACH TO LARGE SCALE SIMULATIONS OF FLUID FLOWS IN FRACTURED MEDIA WITH FINITE ELEMENTS ON NONCONFORMING GRIDS Stefano Berronea, Sandra Pieraccinib and Stefano Scial`oc

Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. [email protected], [email protected] [email protected]

Numerical simulation of subsurface flows is a challenging problem since geometrical complexity and huge dimensions of underground geological reservoirs require a large amount of computational power, and uncertainty in the data on rock properties can only be handled with a stochastic approach and performing a huge number of simula- tions. As a consequence efficiency and large applicability of numerical algorithms is of paramount importance. A novel optimization method to discrete fracture network (DFN) approach for simula- tions of subsurface flows is presented here. DFN models are complex 3D networks of planar polygons resembling underground fractures and, as a first approximation, only flow in the fractures is evaluated, neglecting the percolation through the surrounding rock matrix which occurs on a different temporal scale. A classical approach to the problem consists in the generation of a finite element conforming triangulation of the network and the resolution of the resulting large algebraic system of equations. The optimization approach described in [1, 2, 3], instead, seeks the solution as the mini- mum of a PDE constrained functional. The large scale DFN problem is thus split in a number of quasi-independent smaller problems on the fractures that can be solved in parallel on multi core or GPU based computer architectures. The algorithm does not rely on a conforming triangulation of the fracture system, so that the meshing process can be performed independently on each fracture. This is of paramount importance for DFN simulations, given the difficulties in the generation of a good quality conforming mesh arising from the intricate nature of fracture intersections. The non conforming meshes are handled by means of modified finite element methods. The application of extended finite element methods (XFEM) is fully analysed [4], but also virtual elements (VEM) are tested. Good efficiency for the optimization algorithm is achieved in conjunction with precon- ditioning techniques and multi-grid approaches appear to be very promising to speed up the convergence of the method. Numerical results on complex configurations are provided.

References

[1] , A PDE-constrained optimization formulation for discrete fracture network flows, SIAM Journal on Scientific Computing, 35 (2013), pp. A908–A935.

30-4 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

[2] S. Berrone, S. Pieraccini, and S. Scialo´, On simulations of discrete fracture network flows with an optimization-based extended finite element method, SIAM Journal on Scientific Computing, 35 (2013), pp. B487–B510.

[3] , An xfem optimization approach for large scale simulations of discrete fracture network flows. Submitted for pubblication, 2013.

[4] , The extended finite element method for discrete fracture network simulations. In preparation.

PRESSURE JUMP INTERFACE LAW FOR THE STOKES-DARCY COUPLING: CONFIRMATION BY DIRECT NUMERICAL SIMULATIONS Thomas Carraroa and Christian Gollb

Institute for Applied Mathematics, Heidelberg University, 69120 Heidelberg, Germany. [email protected] [email protected]

We consider slow incompressible viscous flow over a porous bed which is made up of a periodic repetition of a so called ‘unit cell’. The flow is modeled by the steady Stokes equation on the complete domain including the porous part. For computational purposes we upscale the Stokes equation in the porous bed and replace it by the Darcy law. After upscaling, an interface Γ appears between the two domains and relevant boundary conditions at the interface have to be defined. It is well known that on the interface Γ the Beavers-Joseph-Saffmann condition holds true for the effective fluid-velocity ueff :

eff eff bl ∂u1 u1 + εC1 = 0. (BJS) ∂x2

Additionally, after [1, 2], the Darcy pressure in the porous domain pD and the effective pressure in the fluid region peff fulfill a pressure jump law:

eff eff bl ∂u1 pD = p + Cω . (PJL) ∂x2

The subject of this talk is the numerical confirmation of the pressure jump (PJL) by a direct finite element simulation of the flow on the microscopic level. Additionally, we show the numerical results for the confirmation of (BJS). Therefore, we compute the bl bl constants C1 and Cω , which are defined as averages of an appropriate boundary layer problem solved by a finite element approximation. A goal oriented adaptive scheme is applied for the grid refinement allowing a reduction of the computational costs without loss of accuracy. To verify the interface law, we compute the solution of the microscopic problem for different values of ε (the ratio between the length of the unit cell and the height of the

30-5 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs computational domain) and show several convergence results. All the computations are performed for two different inclusions, namely circles and ellipses, since the shape of the pores strongly influences the behavior of the pressure at the interface.

References

[1] Marciniak-Czochra, Anna and Mikelic,´ Andro, Effective Pressure Inter- face Law for Transport Phenomena between an Unconfined Fluid and a Porous Medium Using Homogenization, Multiscale Modeling & Simulation, 10, 285–305, (2012)

[2] Jager,¨ Willi and Mikelic,´ Andro, On the Interface Boundary Condition of Beavers, Joseph, and Saffman, SIAM Journal on Applied Mathematics, 60, 1111– 1127, (2000)

EFFICIENT BAYESIAN UNCERTAINTY QUANTIFICATION OF SUBSURFACE FLOW MODELS USING NESTED SAMPLING AND SPARSE POLYNOMIAL CHAOS SURROGATES Ahmed H. Elsheikh1a,2, Mary F. Wheeler1 and Ibrahim Hoteit2

1 Center for Subsurface Modeling (CSM), Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, TX, USA [email protected] 2 Dept. of Earth Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

An accelerated Bayesian uncertainty quantification method based on the nested sam- pling (NS) algorithm and non-intrusive stochastic collocation method is presented. Nested sampling is an efficient Bayesian sampling algorithm that builds a discrete rep- resentation of the posterior distributions by iteratively re-focusing a set of samples to high likelihood regions. NS allows representing the posterior probability distri- bution function (PDF) with a smaller number of samples and reduces the curse of dimensionality effects. The main difficulty of the nested sampling algorithm is in a constrained sampling step which is commonly performed using a random walk Markov chain Monte-Carlo (MCMC) algorithm. In the current manuscript, we perform a two– stage sampling using a polynomial chaos response surface to filter out rejected samples in the Markov chain Monte-Carlo method. The combined use of nested sampling and the two–stage MCMC based on approximate response surfaces provides significate com- putational gains in terms of the number of simulation runs. The proposed algorithm is applied for calibration and model selection of subsurface flow models.

30-6 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

HIGH-ORDER CUT-CELL TECHNIQUES FOR NUMERICAL UPSCALING IN POROUS MEDIA Christian Engwer

Institute for Computational und Applied Mathematics, University of M¨unster,Germany [email protected]

For simulations of processes in porous media usually a homogenized macroscopic scale is considered. On the macro scale empirical laws are used, which require effective parameters. The models, as well a the parameters are directly linked to pore scale physics and the geometry of the pore space. Pore scale simulations can help where experiments are not possible, or hard to conduct. They give an opportunity for different application, e.g. parameter estimation or model verification. We discuss practical challenges when trying to employ direct pore-scale simulations for numerical upscaling. Mesh-generation problems can be overcome using cut-cell techniques. We present the Unfitted Discontinuous Galerkin method, a higher order cut-cell method. This method allows to solve PDEs on domains with a complicated geometric shape. It uses finite element meshes which are significantly coarser then those required by standard conforming finite element approaches and is flexible enough to be used for elliptic, hyperbolic and parabolic problems. Essential boundary conditions are incorporated weakly. Recent extensions allow to solve coupled bulk-surface problems. Further challenges arise for the linear solver and for parallel computations.

30-7 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

ADJOINTS OF FINITE ELEMENT MODELS Patrick E. Farrell1a,2 and Simon W. Funke1,3

1Department of Earth Science and Engineering, Imperial College London, London, UK [email protected] 2Center for Biomedical Computing, Simula Research Laboratory, Oslo, Norway 3Grantham Institute for Climate Change, Imperial College London, London, UK.

The derivatives of PDE models are key ingredients in many important algorithms of computational mathematics. They find applications in diverse areas such as sensitivity analysis, PDE-constrained optimisation, continuation and bifurcation analysis, error estimation, and generalised stability theory. These derivatives, computed using the so-called tangent linear and adjoint models, have made an enormous impact in certain scientific fields (such as aeronautics, me- teorology, and oceanography). However, their use in other areas has been hampered by the great practical difficulty of the derivation and implementation of tangent lin- ear and adjoint models. In his recent book [U. Naumann, The Art of Differentiating Computer Programs, SIAM, 2011], Naumann describes the problem of the robust au- tomated derivation of parallel tangent linear and adjoint models as “one of the great open problems in the field of high-performance scientific computing”. In this talk, we present an elegant solution to this problem for the common case where the forward model may be written in variational form, and discuss some of its applications.

30-8 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

A GLOBAL JACOBIAN METHOD FOR MORTAR DISCRETIZATIONS OF NONLINEAR POROUS MEDIA FLOWS Benjamin Ganis1a, Mika Juntunen1, Gergina Pencheva1, Mary F. Wheeler1 and Ivan Yotov2

1Center for Subsurface Modeling (CSM), Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, TX, USA [email protected] 2Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

We describe a non-overlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. The new methods are designed to be less complex than a previously developed nonlinear mortar algorithm, which required nested Newton iterations and a forward difference approximation. Furthermore, efficient linear preconditioners can be applied to speed up the iteration. Consequently, we are able to demonstrate greatly improved parallel scalability for large nonlinear systems.

30-9 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

DIRECT NUMERICAL SIMULATION OF TWO-PHASE FLOW AT THE PORE SCALE Ali Q Raeinia, Branko Bijeljicb and Martin J Bluntc

Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK [email protected], [email protected], [email protected]

Direct numerical simulations of two-phase flow at the micron scale help understand and provide vital information on the pore-scale mechanisms controlling two-phase flow in porous media. In this study, we discuss a new sharp surface force model developed for computing capillary forces at the micron scale, in a volume off fluid based finite volume framework. We discuss the difficulties encountered in modelling the high capillary forces at such small scales; and how this new formulation, along with a new filtering scheme to remove non-physical forces from the capillary forces, helps to overcome these difficulties and predict the flow by solving the Navier-Stokes equations. We present exemplar applications of this new method in modelling two-phase flow through simple pure geometries. Particularly, we present simulation results on snap-off and layer flow, in drainage and imbibition, and a methodology to upscale the results to obtain the relevant information required to describe the flow at larger scales. Finally, we present sample simulations for modelling two-phase flow directly on micro-CT images of porous media, and study the effect of capillary number on the macroscopic properties of porous media, such as relative permeability curves and residual non-wetting phase saturations.

30-10 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

MODELING FLOW WITH NONPLANAR FRACTURES Gurpreet Singha, Omar Al-Hinaib, Gergina Penchevac and Mary F. Wheelerd

Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX, USA. [email protected], [email protected], [email protected], [email protected]

Modeling flow with fractures can be challenging due to complex fracture geometries, strong variations in length and time scales as well as the need to combine multiple flow models. In this talk we propose a general methodology for coupling fracture and reservoir flow which is capable of handling multiple physics under the same frame- work while accounting for reservoir complexities. This is accomplished by separate representations of the fracture and reservoir models, followed by a coupling of the two problems using appropriate boundary conditions and forcing functions. A multi-point flux mixed finite element method and mimetic finite difference method are used as the discretization schemes for the reservoir and the fracture, respectively. The methods are locally conservative and allow for accurate flux approximation as well as general hexahedral grids and non-planar fractures. An appropriate choice of convergence cri- teria and numerical stabilization consistent with the physical problem is then utilized to iteratively couple the resulting two flow systems.

COMPUTATIONAL ENVIRONMENTS FOR ENERGY AND ENVIRONMENTAL MODELING IN POROUS MEDIA Mary F. Wheeler

Center for Subsurface Modeling (CSM), Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, TX, USA [email protected]

We discuss a framework for modeling flow, mechanics and chemistry in porous media. Applications include carbon sequestration in saline aquifers, polymer flooding, and geomechanics. Issues include accuracy of discretizations on general grids including distorted hexahedra meshes, solvers and incorporating uncertainty quantification and parameter estimation in the framework.

30-11 30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

MULTISCALE DOMAIN DECOMPOSITION METHODS FOR POROUS MEDIA FLOW COUPLED WITH GEOMECHANICS Ivan Yotov

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA [email protected]

We consider numerical modeling of the system of poroelasticity, which describes fluid flow in deformable porous media. The focus is on locally mass conservative flow dis- cretizations that provide efficient and accurate multiscale approximations on rough grids and for highly heterogeneous media. We employ a multiscale mortar finite el- ement method, where the equations in the coarse elements (or subdomains) are dis- cretized on a fine grid scale, while continuity of normal velocity and stress between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, optimal order convergence is obtained for the method on the fine scale. The algebraic system is re- duced via a non-overlapping domain decomposition to a coarse scale mortar interface problem that is solved efficiently using a multiscale flux basis. This is joint work with Ben Ganis, Bin Wang, and Mary Wheeler (UT Austin), Ruijie Liu (BP), and Gary Xue (Shell).

30-12 31: Mini-Symposium: PDEs on Surfaces

31 Mini-Symposium: PDEs on Surfaces

Organiser: Charlie Elliott

31-1 31: Mini-Symposium: PDEs on Surfaces

UNFITTED FINITE ELEMENT METHODS USING BULK MESHES FOR SURFACE PARTIAL DIFFERENTIAL EQUATIONS Klaus Deckelnick1, Charles M. Elliott2a and Tom Ranner2b

1Institut f¨urAnalysis und Numerik, Otto–von–Guericke–Universit¨atMagdeburg, Universit¨atsplatz2, 39106 Magdeburg, Germany; [email protected] 2Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK; [email protected], [email protected]

In this talk we propose two different unfitted finite element methods for solving an elliptic partial differential equation on a given hypersurface. These methods use a regular triangulation of an ambient domain and perform calculations on an induced triangulation of either the surface or a narrow band around the surface. We present an error analysis for both approaches and show how these methods can be combined in order to solve a parabolic equation on a family of evolving hypersurfaces.

31-2 31: Mini-Symposium: PDEs on Surfaces

DISCONTINUOUS GALERKIN METHODS FOR SURFACE PDES Andreas Dednera, Pravin Madhavanb and Bj¨ornStinner

Mathematics Institute and Centre for Scientific Computing, University of Warwick, Coventry, UK. [email protected], [email protected], [email protected]

Partial differential equations (PDEs) on manifolds have become an active area of re- search in recent years due to the fact that, in many applications, models have to be formulated not on a flat Euclidean domain but on a curved surface. For example, they arise naturally in fluid dynamics and material science but have also emerged in areas as diverse as image processing and cell biology. Finite element methods (FEM) for elliptic problems and their error analysis have been successfully applied to problems on surfaces via the intrinsic approach based on interpolating the surface by a triangulated one. This approach has subsequently been extended to parabolic problems as well as evolving surfaces. However, as in the planar case, there are a number of situations where FEM may not be the appropriate numer- ical method, for instance, advection dominated problems which lead to steep gradients or even discontinuities in the solution. Discontinuous Galerkin (DG) methods are a class of numerical methods that have been successfully applied to hyperbolic, elliptic and parabolic PDEs arising from a wide range of applications. Some of its main advan- tages compared to ‘standard’ finite element methods include the ability of capturing discontinuities as arising in advection dominated problems, and less restriction on grid structure and refinement as well as on the choice of basis functions which makes them ideal for a posteriori error estimation and adaptive refinement. In this talk we will extend the discontinuous Galerkin (DG) framework for a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a priori error estimates by relating the latter to the original surface via a surface lifting operator. The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method. This is then verified numerically for a number of test problems.

31-3 31: Mini-Symposium: PDEs on Surfaces

PATTERN FORMATION IN MORPHOGENESIS ON EVOLVING BIOLOGICAL SURFACES: THEORY, NUMERICS AND APPLICATIONS Anotida Madzvamuse1, Raquel Barreira2, Charles M. Elliott3, Ammon J. Meir4 and Necibe Tuncer5

1University of Sussex, School of Mathematical and Physical Sciences, Pevensey III, 5C15. BN1 9QH. Brighton, UK. [email protected] 2Escola Superior de Tecnologia do Barreiro/IPS, Rua Am´ericoda Silva Marinho-Lavradio, 2839-001 Barreiro, Portugal 3Mathematics Institute and Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, UK 4Department of Mathematics and Statistics, Auburn University, USA 5Department of Mathematics, University of Tulsa, USA

In this talk, I will present our most recent results based on two finite element formula- tions: (i) the surface finite element and (ii) radially projected finite element methods applied to solving partial differential equations of reaction-diffusion type on arbitrary stationary and evolving surfaces. Reaction-diffusion equations on evolving surfaces are formulated using the material transport formula, surface gradients and diffusive conser- vation laws. The evolution of the surface is defined by a material surface velocity. The radially projected finite element method differs from the surface finite element method in that it provides a conforming finite element discretization which is ”logically” rect- angular. However, this property restricts the general applicability of the numerical method to arbitrarily evolving surfaces, a key advantage for the evolving surface finite element method. To demonstrate the capability, flexibility, versatility and generality of the numerical methodologies proposed, I will present various numerical results. This methodology provides a framework for solving partial differential systems on contin- uously evolving domains and surfaces with numerous applications in developmental biology, cancer research, wound healing, tissue regeneration, and cell motility among many others , where reaction-diffusion systems are routinely applied.

31-4 31: Mini-Symposium: PDEs on Surfaces

AN ALE ESFEM FOR SOLVING PDES ON EVOLVING SURFACES Vanessa Styles

Department of Mathematics, University of Sussex, Brighton, England [email protected]

Numerical methods for approximating the solution of partial differential equations on evolving hypersurfaces using surface finite elements on evolving triangulated surfaces are presented. In the ALE ESFEM the vertices of the triangles evolve with a velocity which is normal to the hypersurface whilst having a tangential velocity which is ar- bitrary. This is in contrast to the original evolving surface finite element method in which the nodes move with a material velocity. Numerical experiments are presented which illustrate the value of choosing the arbitrary tangential velocity to improve mesh quality.

31-5 31: Mini-Symposium: PDEs on Surfaces

NUMERICAL SIMULATIONS OF CHEMOTAXIS-DRIVEN PDES ON SURFACES Stefan Tureka and Andriy Sokolovb

Institute of Applied Mathematics, TU Dortmund, Germany [email protected] [email protected]

In the last twenty years one can observe a rapid and consistent growth of interest for bio- mathematical applications. Among them are modeling of tumor invasion and metasta- sis (Chaplain et al.), modeling of vascular network assembly (Preziosi et al.), pattern formations due to the Turing-type instability (Murray et al.) or chemotaxis-driven processes (Horstmann et al.), protein-protein interaction on the membrane (Gory- achev, Bastiaens) and others. These mathematical models, presented as systems of advection-reaction-diffusion equations, can take into consideration various biological processes (e.g. transport, random walk, reaction, chemotaxis, growth and decay, etc.). In our recent research we combine the system of the generalized Keller-Segel model for multi-species with reaction-diffusion equations on (evolving-in-time) surfaces which can be mathematically written in the following form

 species-species interaction species-agents interaction random walk/diffusion z n }| !{ z m }| !{ ∂ui z }| {  X X  = Du∆u + ∇ ·  κ u ∇u − χ u ∇c  + ∂t i i  i,k i k i,k i k   k=1,k6=i k=1  kinetics z }| { + fi(c, ρ), in Ω, (1) decay production zm }| { zn }| { ∂cj X X = Dc∆c − α c + β u +g (u, ρ), in Ω (2) ∂t j j k,j k k,j k j k=1 k=1 diffusion on surface source ∂ρ z }| { z }| { l = Dρ∆ ρ + s (u, c), on Γ(t) (3) ∂t l Γ l l where ui(x, t)(i = 1, n) and cj(x, t)(j = 1, m) are some solutions (for example, species d and chemical agents) defined in a domain Ω ⊂ R (d = 1, 2, 3), and ρl (l = 1, p) are solutions defined on a surface Γ(t) ⊂ Ω. In order to get an accurate numerical solution in a reasonably finite time one has to construct an efficient, fast and robust numerical scheme for a sufficiently large class of partial differential equations. In our talk we investigate FCT and level-set techniques which help to overcome numerical challenges while preforming numerical simulation for the chemotaxis-driven PDEs on evolving-in- time surfaces (1)–(3).

31-6 32: Mini-Symposium: Sensitivity analysis and optimization for fluid-structure interaction problems

32 Mini-Symposium: Sensitivity analysis and op- timization for fluid-structure interaction prob- lems

Organisers: Thomas Richter and Thomas Wick

32-1 32: Mini-Symposium: Sensitivity analysis and optimization for fluid-structure interaction problems

PARAMETER ESTIMATION IN FLUID-STRUCTURE INTERACTION AND SUBSURFACE FLOWS Ahmed H. Elsheikh1, Thomas Richter2, Mary F. Wheeler1 and Thomas Wick1,2a

1 Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA 2 Institute of Applied Mathematics, Heidelberg University, Germany a [email protected]

We present forward and inverse modeling for coupled problems such as fluid-structure interaction and the coupled Biot system with elasticity. Our aim is to estimate material coefficients such as permeability coefficients or Lam´eparameters. In particular, such estimation problems are quite important in subsurface modeling because this is a hot topic in reservoir engineering. Two optimization algorithms are presented. First, standard gradient-based opti- mization (for a stationary setting) showing the robustness when all derivatives are correctly assembled. This is exemplified with the help of a prototypical setting where an elastic structure interacts with a surrounding fluid. Second, we use non-intrusive Bayesian inverse modeling for the coupled, nonstationary Biot-Lam´e-Navier problem. Here, some reservoir (the pay-zone) is modeled as a poroelastic medium with the help of Biot’s equations. A surrounding medium (the non-pay zone) is modeled as a static elastic structure. In both cases, we prefer a monolithic setting of the forward problem. In fact, for gradient-based optimization this a indispensable requirement. We present different examples in two and three dimensions in order to show the performance of our algorithms.

32-2 32: Mini-Symposium: Sensitivity analysis and optimization for fluid-structure interaction problems

TOWARDS OPTIMAL CONTROL OF LARGE DEFORMATION FSI PROBLEMS INCLUDING CONTACT AND TOPOLOGY CHANGE Thomas Richtera and Thomas Wick

Institute of Applied Mathematics, Heidelberg University, Germany [email protected]

Some fluid-structure interaction (FSI) problems involve very large deformation of the elastic structures. A prominent example is the flow of blood in the heart. If the moving heart valves are included in the simulation, the structure will even undergo self-contact which can lead to topology change (if the heart chamber is closing). This contribution has two parts. First, we will introduce the fully Eulerian formula- tion as a monolithic model for FSI problems. This formulation can be regarded as the antipode to the Arbitrary Lagrangian Eulerian (ALE) formulation for FSI problems. While the ALE method uses fixed reference domains (Lagrangian in the solid) for mod- eling both subproblems, the Eulerian formulation uses a natural Eulerian framework for both parts. This construction allows for the simulation of problems with very large deformation and contact. Second, we discuss gradient based optimization techniques for FSI problems. The basic concepts will be introduced using the ALE formulation. It will be easy to see, that this ALE formulation is not able to handle problems with large deformation or even contact. Finally, we will describe first steps for an extension of gradient based optimization techniques to the Eulerian formulation.

CALCULATION OF SENSITIVITIES FOR FLUID-STRUCTURE INTERACTIONS Thomas Wick1 and Winnifried Wollner2

1 The Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, USA [email protected] 2University of Hamburg, Department of Mathematics, Bundesstr 55, 20146 Hamburg, Germany [email protected]

In this talk we will consider a stationary model for fluid structure interaction of an incompressible fluid with an elastic structure using the ALE-framework. In particular we will discuss differentiability of the solution operator of the problem.

32-3 33: Mini-Symposium: Stochastic finite elements and PDEs

33 Mini-Symposium: Stochastic finite elements and PDEs

Organiser: Max Gunzburger

33-1 33: Mini-Symposium: Stochastic finite elements and PDEs

A POSTERIORI ERROR ESTIMATION FOR STOCHASTIC GALERKIN FEMS Alex Bespalov1,2a, Catherine Powell2b and David Silvester2c

1 School of Mathematics, University of Birmingham, Birmingham, UK, 2 School of Mathematics, University of Manchester, Manchester, UK [email protected] [email protected], [email protected]

Stochastic Galerkin finite element approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. Given a parametrisation of the data in terms of a large, possibly infinite, number of random variables, this approach allows the original PDE problem to be reformulated as a para- metric, deterministic PDE on a parameter space of high, possibly infinite, dimension. A typical strategy is to combine conventional (h-) finite element approximation on the spatial domain with spectral (p-) approximation on a finite-dimensional manifold in the (stochastic) parameter space. For approximations relying on low-dimensional manifolds in the parameter space, stochastic Galerkin finite element methods have superior convergence properties to standard sampling techniques. On the other hand, the desire to incorporate more and more parameters (random variables) together with the need to use high-order polyno- mial approximations in these parameters inevitably generates very high-dimensional discretised systems. This in turn means that adaptive algorithms are needed to ef- ficiently construct approximations, and fast and robust linear algebra techniques are essential for solving the discretised problems. Both strands will be discussed in the talk. We outline the issues involved in a poste- riori error analysis of computed solutions and present a practical a posteriori estimator for the approximation error. We introduce a novel energy error estimator that uses a parameter-free part of the underlying differential operator—this effectively exploits the tensor product structure of the approximation space and simplifies the linear algebra. We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the discrete space and establish two-sided estimates of the er- ror reduction for the corresponding enhanced approximations. These give computable estimates of the error reduction that depend only on the problem data and the original approximation. We show numerically that these estimates can be used to choose the enrichment strategy that reduces the error most efficiently. This work is supported by the EPSRC grant EP/H021205/1.

33-2 33: Mini-Symposium: Stochastic finite elements and PDEs

WEAK TRUNCATION ERROR ESTIMATES FOR ELLIPTIC PDES WITH LOGNORMAL COEFFICIENTS Julia Charrier1 and Arnaud Debussche2

1 LATP, Aix Marseille universit´e,13453 Marseille, France. [email protected] 2 ENS Cachan Bretagne, 35170 Bruz, France. [email protected]

In the context of uncertainty quantification, uncertainties on some physical properties of a medium can be modeled by using random fields, leading to partial differential equations with random coefficients. We are here concerned with the case of elliptic partial differential equations with lognormal coefficients. More precisely, we consider the following equation: for almost all ω

 −div(a(ω, x)∇u(ω, x)) = f(x) on D u(ω, x) = 0 on ∂D, where a(ω, x) = eg(ω,x) and g is a gaussian field. We include in particular in our study cases where the lognormal coefficient a may have realizations with low regularity (typically only H¨oldercontinuous). The aim is then to compute the law of the solution u. Such models are in particular widely used in hydrogeology to model flow in porous media with uncertainty. Severeral numerical methods for such equations (such as stochastic galerkin or stochastic collocation methods among others) are based on the approximation of the random coefficient a by a finite number N of random variables. The numerical cost of these methods tends generally to increase drastically with the number N of random variables used, hence it is important to get sharp estimates of the error resulting on the solution. In this talk we present weak error bounds, in the case where the approximation of the coefficient a is performed through a truncation of the Karhunen-Lo`eve expansion of the gaussian field g. More precisely we give bounds in C1 norm of the error on the law of the solution u resulting from the approximation of the coefficient a. We get a weak order of convergence equal to twice the strong order of convergence and illustrate numerically the optimality of the order of convergence obtained. These results (which can be found in [2]) complete and improve the results of [1]. Moroever we complement these results by providing the first term of the asymptotic expansion of the weak truncation error in a particular case, the aim of studying such asymptotic expansion being to justify the use of Richardson extrapolation with respect to the truncation order N.

References

[1] Charrier J. - Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients, SIAM Journal on Numerical Analysis, 2012.

33-3 33: Mini-Symposium: Stochastic finite elements and PDEs

[2] Charrier J., Debussche A. - Weak truncation error estimates for elliptic PDEs with lognor- mal coefficients, Stochastic Partial Differential Equations : Analysis and Computations, 2013

ADAPTIVE ANISOTROPIC SPECTRAL STOCHASTIC METHODS FOR UNCERTAIN SCALAR CONSERVATION LAWS Alexandre Ern1, Olivier Le Maˆıtre2 and Julie Tryoen1,3

1Universit´eParis-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne la Vall´eecedex 2, France [email protected] 2LIMSI-CNRS, UPR-3251, Orsay, France [email protected] 3INRIA Bordeaux Sud-Ouest, Bacchus Team, 33405 Talence cedex 1, France [email protected]

We develop adaptive anisotropic discretization schemes for conservation laws with stochastic parameters. A Finite Volume scheme is used for the deterministic discretiza- tion, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with Roe-type solver, see [1]. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in the space and time so that the constructed stochastic approximation space depends on space and time. The dynami- cally evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis. Efficiency of the overall adaptive scheme is assessed on the stochastic traffic equation with uncertain initial conditions and velocity leading to ex- pansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in view of dealing with problems involving a larger number of stochastic parameters. More details on the methodology and the numerical results can be found in [2].

References

[1] J. Tryoen, O. Le Maˆıtre, M. Ndjinga and A. Ern, Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems, J. Comput. Phys., 229, 6485–6511 (2010).

[2] J. Tryoen, O. Le Maˆıtre, and A. Ern, Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws, SIAM J. Sci. Comput., 34(5), A2459–A2481 (2012).

33-4 33: Mini-Symposium: Stochastic finite elements and PDEs

SPARSE ADAPTIVE TENSOR GALERKIN APPROXIMATIONS OF STOCHASTIC PDE-CONSTRAINED CONTROL PROBLEMS Angela Kunoth

Institut f¨urMathematik, Universit¨atPaderborn, Warburger Str. 100, 33098 Paderborn, Germany [email protected]

For control problems constrained by linear elliptic or parabolic PDEs depending on countably many parameters, i.e., on σj with j ∈ N, we prove analytic parameter dependence of the state, the co-state and the control. Moreover, we establish that these functions allow expansions in terms of sparse tensorized generalized polynomial chaos (gpc) bases. Their sparsity is quantified in terms of p-summability of the coefficient sequences for some 0 < p ≤ 1. Resulting a-priori estimates establish the existence of an index set Λ, allowing for concurrent approximations of state, co-state and control for which the gpc approximations attain rates of best N-term approximation. These results serve as the analytical foundation for the development of correspond- ing sparse realizations in terms of deterministic adaptive Galerkin approximations of state, co-state and control on the entire, possibly infinite-dimensional parameter space. The results were obtained jointly with Christoph Schwab (ETH Z¨urich).

FINITE ELEMENT APPROXIMATION OF THE CAHN-HILLIARD-COOK EQUATION Stig Larsson

Department of Mathematical Sciences, Chalmers University of Technology, SE–412 96 Gothenburg, Sweden [email protected]

We study the Cahn-Hilliard-Cook equation, i.e., the Cahn-Hilliard equation perturbed by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate. This is joint work with M. Kov´acsand A. Mesforush.

33-5 33: Mini-Symposium: Stochastic finite elements and PDEs

EXPLORING EMERGING MANYCORE ARCHITECTURES FOR UNCERTAINTY QUANTIFICATION THROUGH EMBEDDED STOCHASTIC GALERKIN METHODS Eric Phipps1, H. Carter Edwards2, Jonathan Hu3 and Jakob T. Ostien4

1Sandia National Laboratories, Optimization and Uncertainty Quantification Department, Albuquerque, NM, USA , [email protected], 2Sandia National Laboratories, Multiphysics Simulation Technologies Department, Albuquerque, NM, USA, [email protected], 3Sandia National Laboratories, Scalable Algorithms Department, Livermore, CA, USA, [email protected], 4Sandia National Laboratories, Mechanics of Materials Department, Livermore, CA, USA, [email protected]

We explore approaches for improving the performance of intrusive or embedded stochas- tic Galerkin uncertainty quantification methods on emerging computational architec- tures. Our work is motivated by the trend of increasing disparity between floating- point throughput and memory access speed on emerging architectures, thus requiring the design of new algorithms with memory access patterns more commensurate with computer architecture capabilities. We first compare the traditional approach for im- plementing stochastic Galerkin methods to non-intrusive spectral projection methods employing high-dimensional sparse quadratures on relevant problems from computa- tional mechanics, and demonstrate the performance of stochastic Galerkin is reason- able. Several reorganizations of the algorithm with improved memory access patterns are described and their performance measured on contemporary manycore architec- tures. We demonstrate these reorganizations can lead to improved performance for matrix-vector products needed by iterative linear system solvers, and highlight further algorithm research that might lead to even greater performance.

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

33-6 33: Mini-Symposium: Stochastic finite elements and PDEs

ADAPTIVE, SPARSE QUADRATURES FOR BAYESIAN INVERSE PROBLEMS Claudia Schillingsa and Christoph Schwabb

Seminar for Applied Mathematics, ETH Zurich, Switzerland. [email protected], [email protected]

Based on sparsity results of a parametric, deterministic formulation for Bayesian in- verse problems, we will propose a class of adaptive, deterministic sparse tensor Smolyak quadrature schemes for the efficient approximate numerical evaluation of expectations under the posterior, given data. Convergence rates for the quadrature approximation are shown, both theoretically and computationally, to depend only on the sparsity class of the unknown and in particular, are provably higher than those of Monte-Carlo (MC) and Markov-Chain Monte-Carlo methods. Problem classes considered are PDEs and large ODE systems with uncertain coefficients and parameters. This work is supported by the European Research Council under FP7 Grant AdG247277.

MULTILEVEL MARKOV CHAIN MONTE CARLO ALGORITHMS FOR UNCERTAINTY QUANTIFICATION IN SUBSURFACE FLOW Aretha Teckentrup

Department of Mathematical Sciences, University of Bath, Bath, BA2 3JU, UK. [email protected]

The quantification of uncertainty in groundwater flow plays a central role in the safety assessment of radioactive waste disposal and of CO2 capture and storage underground. Stochastic modelling of data uncertainties in the rock permeabilities lead to elliptic PDEs with random coefficients. Typical models used for the random coefficients, such as log-normal random fields with exponential covariance, are unbounded and have only limited spatial regularity, making practical computations very expensive and the rigorous numerical analysis challenging. To overcome the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo (MCMC) methods, we develop and analyse a new multilevel MCMC algorithm, based on a hierarchy of spatial levels/grids. We will demonstrate on a typical model problem the significant gains with respect to conventional MCMC that are possible with this new approach, and provide a full convergence analysis of the new algorithm.

33-7 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

34 Mini-Symposium: Superconvergence in DG: anal- ysis and recovery

Organisers: Lilia Krivodonova and Jennifer Ryan

34-1 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

A NEW LAX-WENDROFF DISCONTINUOUS GALERKIN METHOD WITH SUPERCONVERGENCE Wei Guo1, Jianxian Qiu2 and Jing-Mei Qiu3

1 Department of Mathematics, University of Houston, Houston, USA. [email protected] 2 School of Mathematical Sciences, Xiamen University, China. [email protected] 3 Department of Mathematics, University of Houston, Houston, USA. [email protected]

Various superconvergence properties of discontinuous Galerkin (DG) and local DG (LDG) methods for linear hyperbolic and parabolic equations have been investigated in the past. Due to these superconvergence properties, DG and LDG methods have been known to provide good wave resolution properties, especially for long time inte- grations. In this work, we investigate the super convergence property of DG coupled with Lax-Wendroff time discretization. We develop a new Lax-Wendroff (LW) DG method for hyperbolic conservation laws for its super convergence in terms of negative norm, post-processed solutions and extra long time behavior. Especially, we propose to use the local DG method in approximating higher order spatial derivatives in the LW approach. As a results, 2k + 1-th order of convergence is observed for the post processed LWDG solution. The eigen-structure of the new Lax-Wendroff DG scheme is analyzed symbolically; and the 2k + 1-th order of convergence is observed for the physically relevant eigenvalues. A collection of numerical examples for linear equations are presented to verify our observations.

ENERGY NORM ERROR ESTIMATION FOR AVERAGED DISCONTINUOUS GALERKIN METHODS IN ONE SPACE DIMENSION Ferenc Izs´ak

Institute of Mathematics, E¨otv¨osLor´andUniversity, Budapest, Hungary. [email protected]

In the numerical solution of elliptic boundary value problems, a natural measure of the error is the H1-(semi)norm. This does not make sense in case of discontinuous approximations. In the talk, we present an error estimate for the local average of the interior penalty method in the H1-seminorm in one space dimension. Instead of the well-known post-processing we immediately compute the averaged approximation. It also gives a new viewpoint: the interior penalty method can be derived as a lower order modification of a conforming method.

34-2 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) FILTERING: PRACTICAL CONSIDERATIONS WHEN APPLIED TO VISUALIZATION Robert M. Kirby

School of Computing, University of Utah, USA [email protected]

The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. Although one of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to combine high-order discretizations on an inter-element level while allowing discontinuities between elements, this flexibility generates a plethora of difficulties when one attempts to post-process DG fields for analysis and evaluation of scientific results. Smoothness-increasing accuracy-conserving (SIAC) filtering enhances the smoothness of the field by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. In this talk, we will present our efforts as pertains to attempting to apply SIAC filtering to DG fields for the purposes of visualization. Included in the topics to be discussed will be comparisons between exact and approximate quadrature algorithms [1], extensions of SIAC filtering to triangular meshes [2, 3] and implementation and parallelization details [4]. This is joint work with the PhD work of Dr. Hanieh Mirzaee (School of Computing, University of Utah).

References

[1] Hanieh Mirzaee, Jennifer K. Ryan and Robert M. Kirby,“Quantification of Errors Introduced in the Numerical Approximation and Implementation of Smoothness- Increasing Accuracy Conserving (SIAC) Filtering of Discontinuous Galerkin (DG) Fields”, Journal of Scientific Computing, 45, 447–470, (2010).

[2] Hanieh Mirzaee, Liang Yue, Jennifer K. Ryan and Robert M. Kirby,“Smoothness- Increasing Accuracy-Conserving Post-Processing (SIAC) Postprocessing for Dis- continuous Galerkin Solutions Over Structured Triangular Meshes”, SIAM Journal of Numerical Analysis, Vol. 49, No. 5, pages 1899-1920, 2011.

[3] Hanieh Mirzaee, James King, Jennifer K. Ryan and Robert M. Kirby,“Smoothness- Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solu- tions Over Unstructured Triangular Meshes”, SIAM Journal of Scientific Com- puting, Vol. 35, No. 1, pages 212-230, 2013.

[4] Hanieh Mirzaee, Jennifer K. Ryan and Robert M. Kirby, “Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions”, Journal of Scientific Computing, Vol. 52, No. 1, pages 85-112, 2012.

34-3 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

ERROR ESTIMATION FOR THE DISCONTINOUS GALERKIN METHOD APPLIED TO HYPERBOLIC CONSERVATION LAWS Lilia Krivodonovaa and Noel Chalmersb

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada. [email protected], [email protected]

Error estimation for hyperbolic problems presents theoretical and computational dif- ficulties due to discontinuities that such equations develop. Additionally, the popular adjoint problem approach to computing an estimate of the error is not very suitable for transient hyperbolic equations. We derive a partial differential equation that describes propagation of the exact error for one-dimensional hyperbolic problems. We solve this equation numerically to obtain accurate error estimates even in the presence of discontinuities. We dis- cuss the connection of the error governing equation with the superconvergence at the Radau points. We then show that the structure of the error is more complex than superconvergence theory would allow and that the proposed estimator works where superconvergence based estimators fail.

34-4 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

COMPUTATIONALLY EFFICIENT BOUNDARY FILTERING USING SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) METHODS Xiaozhou Li

Delft Institute of Applied Mathematics, Delft University of Technology 2628 CD Delft, The Netherlands. [email protected]

The discontinuous Galerkin (DG) method continues to be a popular method for many scientific areas such as fluid-structure interaction, two phase flow, etc. By investigating the superconvergence properties of a DG approximation, it is possible to obtain a higher order accurate solution through Smoothness-Increasing Accuracy-Conserving Filtering by post-processing as outlined in [1, 2]. This post-processing technique relies on the negative-order norm estimates of DG approximation and can filter out oscillations in the error and maintain or improve the accuracy. However, extending this technique to non-periodic boundary conditions is challenging. In this talk, we discuss creating a boundary filter that both improves accuracy and smoothness and is furthermore computationally efficient. We address improving the one-sided problem by introducing general B-splines near the boundary region which is more efficient. For the discontinuous Galerkin method, we show that this one-sided post-processing technique increases the smoothness and conserves the accuracy not only for uniform mesh but also for non-uniform mesh. This is joint work with Jennifer Ryan (University of East Anglia), Mike Kirby (University of Utah) and Kees Vuik (Delft University of Technology).

References

[1] B. Cockburn, M. Luskin, C.-W. Shu, E. S¨uli, Enhanced accuracy by post- processing for finite element methods for hyperbolic equations, Mathematics of Computation, 72 (2003), pp.577-606.

[2] M. Steffan, S. Curtis, R.M. Kirby, and J.K. Ryan, Investigation of smoothness enhancing accuracy-conserving filters for improving streamline integration through discontinuous fields, IEEE-TVCG, 14 (2008), pp. 680-692.

[3] X. Li, J.K. Ryan, R.M. Kirby and C. Vuik, Computationally efficient position- dependent smoothness-increasing accuracy-conserving (SIAC) filtering: the uni- form mesh case, Submitted.

34-5 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

SUPERCONVERGENCE OF A HDG METHOD FOR FRACTIONAL DIFFUSION PROBLEMS Kassem Mustapha1 and Bernardo Cockburn2

1Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Saudi Arabia [email protected] 2School of Mathematics, University of Minnesota, USA [email protected]

In this talk, we propose a spatial hybridizable discontinuous Galerkin (HDG) method for the numerical solution of fractional sub-diffusion problems of the form ∂tu − −α 2 ∂t ∇ u = f on (0,T ) × Ω with −1 < α < 0. Noting that, as α → 0, our model problem becomes u0 − ∇2u = f, which is just the classical heat equation. For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if the HDG approximations are taken to be piece-
wise polynomials of degree r ≥ 0, the approximations to u in the L∞ 0,T ; L2(Ω) -norm
r+1 and to −∇u in the L∞ 0,T ; L2(Ω) -norm are proven to converge with the rate h , where h is the maximum diameter of the elements of the spatial mesh. Moreover, for r ≥ 1 and by using quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate faster than plog(T h−2/(α+1)) hr+2. Some numerical examples will be given at the end of the presentation.

34-6 34: Mini-Symposium: Superconvergence in DG: analysis and recovery

POST-PROCESSING DISCONTINUOUS GALERKIN SOLUTIONS TO VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS: ANALYSIS AND SIMULATIONS Jennifer K. Ryan1 and Kassem Mustapha2

1School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK [email protected] 2Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia [email protected]

In this talk we discuss superconvergence extraction for Volterra integro-differential equations with smooth and non-smooth kernels. Specifically, we discuss viable super convergence extraction techniques via a post-processed discontinuous Galerkin (DG) method. Using nodal interpolation, a global superconvergence error bound (in the L∞- norm) is established. For a non-smooth kernel, a family of non-uniform time meshes is used to compensate for the singular behaviour of the exact solution near t = 0. The derived theoretical results are numerically validated in a sample of test problems, demonstrating higher-than-expected convergence rates.

34-7 35: Mini-Symposium: Time-Domain Boundary Integral Equations

35 Mini-Symposium: Time-Domain Boundary In- tegral Equations

Organisers: Lehel Banjai and Stefan Sauter

35-1 35: Mini-Symposium: Time-Domain Boundary Integral Equations

A FULLY DISCRETE KIRCHHOFF FORMULA BASED ON CQ AND GALERKIN BEM Lehel Banjai1, Antonio Laliena2 and Francisco-Javier Sayas3

1 School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK [email protected] 2Dep. Matem´aticas,EUPLA, Universidad de Zaragoza, La Almunia, Spain [email protected] 3Department of Mathematical Sciences, University of Delaware, Newark DE, USA [email protected]

In this talk we present some novel analytical techniques to study all discretization errors for a fully discrete approximation of the Kirchhoff Boundary Integral Equation using: Galerkin BEM, data interpolation, and Convolution Quadrature in the time domain. The most delicate part of this analysis is related to the Galerkin semidiscretization-in- space. This analysis is carried out by studying a dynamical system that is equivalent to the set of semidiscrete equations. This entire process is developed without any re- course to the Laplace transform of the discrete operator, thus providing tight estimates with well controled constants for long time simulations. Finally, time discretization is studied in a similar way, transfering well known tools for analysis of discretizations of the wave equation on bounded sets to exotic transmission problems.

35-2 35: Mini-Symposium: Time-Domain Boundary Integral Equations

TIME-DOMAIN FEM/BEM COUPLING Lehel Banjai1, Volker Gruhne2, Christian Lubich3 and Francisco-Javier Sayas4

1 Heriot-Watt University, Edinburgh, UK. [email protected] 2 Max Planck Institute for Mathetics in the Sciences, Leipzig, Germany. 3 University of T¨ubingen, Germany. 4 University of Delaware, USA.

We will discuss the numerical simulation of acoustic wave propagation with localized inhomogeneities. To do this we will apply a standard finite element method (FEM) in space and explicit time-stepping in time on a finite spatial domain containing the in- homogeneities. The equations in the exterior computational domain will be dealt with a time-domain boundary integral formulation discretized by the Galerkin boundary element method (BEM) in space and convolution quadrature in time. We will give the analysis of the proposed method, starting with the proof of a positivity preservation property of convolution quadrature as a consequence of a variant of the Herglotz the- orem. Combining this result with standard energy analysis of leap-frog discretization of the interior equations will give us both stability and convergence of the method. Numerical results will also be given.

35-3 35: Mini-Symposium: Time-Domain Boundary Integral Equations

CONVOLUTION-IN-TIME APPROXIMATIONS OF TDBIES Penny J Davies1 and Dugald B Duncan2

1Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UK [email protected] 2Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University, Edinburgh, UK [email protected]

We present a new framework for the temporal approximation of TDBIEs which can be combined with either collocation or Galerkin approximation in space. It shares some properties of convolution quadrature, but instead of being based on an underlying ODE solver the approximation is explicitly constructed in terms of basis functions which have compact support, and hence has sparse system matrices. The time-stepping method is derived as an approximation for convolution Volterra integral equations (VIEs): at time step tn = nh the VIE solution is approximated in a Pn “backward time” manner in terms of basis functions φj by u(tn −t) ≈ j=0 un−jφj(t/h) for t ∈ [0, tn]. When the basis functions are cubic B-splines with the “parabolic runout” conditions at t = 0 the method is fourth order accurate, and numerical test results indicate that it gives a very stable approximation scheme for acoustic scattering TDBIE problems.

35-4 35: Mini-Symposium: Time-Domain Boundary Integral Equations

QUADRATURE SCHEMES AND ADAPTIVITY FOR 2D TIME DOMAIN BOUNDARY ELEMENT METHODS (TD-BEM) Matthias Gl¨afkea and Matthias Maischakb

BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge, U.K. [email protected], [email protected]

In this talk we study the transient scattering of acoustic waves by an obstacle in an infinite two dimensional domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to finding the unknown density of the time domain boundary layer operators on the obstacle’s boundary, subject to the boundary data of the known incident wave. Using a Galerkin approach, the unknown density is approximated by a piecewise polynomial function, the coefficients of which can be found by solving a linear system. The entries of the system matrix of this linear system involve, for the case of the two dimensional scattering problem under consideration, integrals over four dimensional space-time manifolds. An accurate computation of these integrals is crucial for the stability of this method. Using piecewise polynomials of low order, the two tempo- ral integrals can be evaluated analytically, leading to kernel functions for the spatial integrals with complicated domains of piecewise support. These spatial kernel functions can be generalised into a class of admissible kernel functions which, as we prove, belong to countably normed spaces [1]. Therefore, a quadrature scheme for the approximation of the two dimensional spatial integrals with admissible kernel functions converges exponentially [3]. Similar results for the three dimensional case can be found in [2, 4]. We also present a fully adaptive scheme that modifies both the spatial and the temporal mesh according to an error estimator, and show numerical experiments un- derlining the theoretical results [1].

References

[1] M. Gl¨afke. Adaptive Methods for Time Domain Boundary Integral Equations. PhD Thesis, Brunel University, 2013.

[2] E. Ostermann. Numerical Methods for Space-Time Variational Formulations of Retarded Potential Boundary Integral Equations. PhD Thesis, Institut f¨urAnge- wandte Mathematik, Leibniz Universit¨atHannover, 2010.

[3] C. Schwab. Variable order composite quadrature of singular and nearly singular integrals. Computing 53, 2 (1994), 173–194.

[4] E. P. Stephan, M. Maischak, E. Ostermann. Transient boundary element method and numerical evaluation of retarded potentials. In Computational Science – ICCS 2008, Vol. 5102 of Lecture Notes in Computational Science , Springer, 2008, 321– 330.

35-5 35: Mini-Symposium: Time-Domain Boundary Integral Equations

SOLVING THE HEAT EQUATION WITH A FAST MULTIPOLE GALERKIN BOUNDARY ELEMENT METHOD Michael Messner1,a, Johannes Tausch2, Martin Schanz1,c and Wolfgang Weiss3

1Institute of Applied Mechanics, Graz University of Technology, Graz, Austria [email protected], [email protected], 2Mathematics Department, Southern Methodist University, Dallas, Texas, USA [email protected], 3Institute of Tools and Forming, Graz University of Technology, Graz, Austria [email protected]

We are solving boundary integral equations related to the transient heat equation [1]. A straight forward space-time Galerkin discretization with equidistant time steps leads to a lower Toeplitz structure in time with dense spatial contributions. The cost of solving such a system by forward substitution is O(N 2M 2) with N being the number of time steps and M the number of spatial unknowns. We accelerate the spatio- temporal farfield by the parabolic fast multipole method [2, 3] to reduce the overall complexity to almost O(NM). On some benchmark examples we show that we achieve the correct convergence rates for a variety of pure and mixed initial boundary value problems. Moreover, we present some results from the thermal simulation of industrial applications.

References

[1] M. Costabel, Boundary Integral Operators for the Heat Equation. Integral Equa- tions and Operator Theory 13:498-552, 1990.

[2] J. Tausch, A fast method for solving the heat equation by layer potentials. Journal of Computational Physics 224(2):956-969, 2007.

[3] M. Messner and J. Tausch and M. Schanz, Fast Galerkin Mehtod for Parabolic Space-Time Boundary Integral Equations. submitted to: Journal of Computational Physics

35-6 35: Mini-Symposium: Time-Domain Boundary Integral Equations

A GENERALIZED CONVOLUTION QUADRATURE WITH VARIABLE TIME STEPPING Stefan A. Sauter

Institut f¨urMathematik, Universit¨atZ¨urich, Winterthurerstr 190, CH-8057 Z¨urich, Switzerland [email protected]

We will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to non-uniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory we will consider the wave equation in exterior domains, which is formulated as a retarded boundary integral equation. This work is in collaboration with Maria Lopez-Fernandez.

ADAPTIVE METHODS FOR RETARDED BOUNDARY INTEGRAL EQUATIONS Stefan A. Sautera and Alexander Veitb

Institute of Mathematics, University of Zurich, Switzerland. [email protected], [email protected]

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the accurate computation of the system matrix entries is the major bottleneck. In order to simplify the arising quadrature problem we use globally smooth and compactly supported basis functions for the time discretization. This furthermore easily allows the use of a variable time-stepping and a variable order of the approximation in time. In order to obtain a scheme that automatically adapts the time grid to local irregu- larities in the solution we use suitable a posteriori error estimators. Various numerical experiments show the behavior of the adaptive algorithm.

35-7 35: Mini-Symposium: Time-Domain Boundary Integral Equations

BEM FOR PARABOLIC PHASE PHANGE PROBLEMS WITH MOVING INTERFACES Johannes Tauscha and Elizabeth Caseb

Department of Mathematics, Southern Methodist University, Dallas, TX, USA. [email protected], [email protected]

Many phase change problems are governed by the heat equation in the liquid and solid domain and the Stefan condition ∂u ∂u v = k s − k l , n s ∂n l ∂n on the boundary between the two. Here, vn is the normal velocity of the interface, u is the temperature, k is a non-dimensionalized diffusion constant, and subscripts in- dicate solid and liquid phase. The goal is to determine the evolution of the unknown interface. We consider a boundary integral formulation based on the Green’s repre- sentation formula of the heat equation Unlike the more standard approaches to this type of problem there are only unknowns on the boundary. However, every time step involves a convolution over the entire history of the problem. The time-dependent integral equation is discretized with the Nystr¨ommethod of [J. Tausch, Applied Num. Math.59, pp. 2843-2856 (2009)]. Here, special attention must be given to extra terms in the Green’s representation formula and the discrete operators due to the moving interfaces. We obtain a simple time-stepping method and illustrate the stability and convergence on a number of integral formulations of solidification problems.

35-8 35: Mini-Symposium: Time-Domain Boundary Integral Equations

A HYBRID APPROACH TO THE TIME MARCHING SOLUTION OF MAXWELL’S EQUATIONS

Daniel S. Weile

Department of Electrical and Computer Engineering, Newark, DE, USA [email protected]

The solution of the transient boundary integral formulation of electromagnetics gen- erally proceeds by one of two methods based on the temporal discretization of the equation. In the convolute quadrature (CQ) approach, the equation is discretized by a mapping from the continuous Laplace domain into the discrete frequency domain and then inverse transformed. On the other hand, the temporal Galerkin (TG) approach works by approximating the temporal dependence of the unknown current as a linear combination of basis functions, and then forcing the projection of the approximation on a given testing space to vanish. The great advantage of CQ relative to TG arises from its predictable convergence properties and the ease of computing the convolution coefficients it requires. Unfortu- nately, another predictable outcome of the CQ procedure is an unphysical numerical dispersion, which can alter the appearance of important time domain effects such as coherence and pulse shaping. Because time delays are computed explicitly in TG, numerical dispersion is avoided. The exact computation of the time delays, however, complicates the evaluation of the convolution coefficients because of the bizarre region shapes that arise in the electromagnetic interaction of patches. In this talk, we introduce a hybrid CQ/TG approach that combines the stability of CQ with the dispersion properties of TG. Specifically, CQ is used to compute the interactions between proximal patches, and TG is used for laches at a greater dis- tance. Numerical results will demonstrate that the stability of CQ is preserved and its dispersion is mitigated.

35-9 35: Mini-Symposium: Time-Domain Boundary Integral Equations

USING SPACE-TIME GALERKIN STABILITY THEORY TO DEFINE A ROBUST COLLOCATION METHOD FOR TIME-DOMAIN BOUNDARY INTEGRAL EQUATIONS IN ELECTROMAGNETICS Elwin van ’t Wout1a, Duncan R. van der Heul2b, Harmen van der Ven1c and Kees Vuik2d

1National Aerospace Laboratory NLR, Amsterdam, Netherlands, [email protected], [email protected] 2Delft Institute of Applied Mathematics, Delft University of Technology, Delft, Netherlands, [email protected], [email protected]

The time domain boundary integral equation (TDIE) method is an effective compu- tational method for modelling electromagnetic scattering phenomena. Because of the formulation in time domain, scatterers with a nonlinear response can be simulated for incident fields with a large frequency band. As a boundary element method, the number of spatial degrees of freedom scales quadratically with the electrical size of the scatterer. These advantages make it a promising method for industrial use in stealth technology and target recognition of electrically large structures. Two of the most widely used discretisation schemes for TDIE methods are space- time Galerkin and collocation in time, also called Marching-on-in-Time (MoT). Both have distinct advantages. The design of space-time Galerkin schemes has been based on a functional analysis of the variational formulation. Coerciveness and stability have been proven for specific Sobolev spaces. The efficiency of MoT schemes has been improved tremendously with the use of accelerators based on plane-wave expansions and fast Fourier transforms. This has led to the successful application of the MoT scheme to electrically large and penetrable structures. Aim of this paper is to combine the two numerical schemes into one method that possesses the advantages of both. That is, the stability theory of the space-time Galerkin method will be used to define a robust collocation scheme. To this end, the functional analytic framework of the space-time Galerkin method will be adapted for different versions of the Electric Field Integral Equation (EFIE). This results in infinite dimensional Sobolev spaces for which the variational formulations are stable. On a discrete level, the two methods will be shown to be equivalent for specific choices of test and basis functions. So for a given space-time Galerkin scheme, a discretely equivalent MoT scheme can be derived. The test and basis functions of the space- time Galerkin scheme will be chosen as elements of specific Sobolev spaces. Then, the equivalencies advocate the use of quadratic spline basis functions in the MoT scheme. Computational experiments confirm the robustness of the MoT scheme when the basis functions in time are chosen according to the mathematical framework of space- time Galerkin schemes. It is anticipated that with the combined experiences from space-time Galerkin and collocation schemes, a TDIE method can be formulated that is sufficiently robust for industrial applications.

35-10 Author index Abdul-Rahman Razi, 2-3 Bijeljic Branko, 30-10 Abert Claas, 6-2 Birch Malcolm J., 22-2, 22-4 Abgrall R., 15-2 Blatt Markus, 20-4 Acosta Gabriel, 14-2 Blunt Martin J, 30-10 Ahmed Naveed, 2-4 B¨ohmerKlaus, 29-2 Ainsworth Mark, 11-2, 17-3, 17-25, 23-2, Boffi Daniele, 2-11 23-3 Bokhove Onno, 8-5 Al-Hinai Omar, 30-11 Bonito Andrea, 26-3 Allendes Alejandro, 11-2 Bordas St´ephaneP.A., 1-2 Antonietti Paola F., 7-2 Bramble James H., 25-2 Apel Thomas, 3-2–3-4, 14-3 Brenner Susanne C., 1-1 Appel¨oDaniel, 17-11 Brewin Mark P., 22-2, 22-4 Armentano Mar´ıaGabriela, 14-2 Brezzi Franco, 18-2, 18-5 Arndt Daniel, 11-3 Bruckner Florian, 6-3 Arrar´asAndr´es,30-3 Brzobohat´yTom´aˇs,28-2, 28-6 Arridge S.R., 4-2 Buffa Annalisa, 1-4, 15-3, 15-6, 17-4 Artina Marco, 2-5 Bugert Beatrice, 4-3 Aschemann Harald, 19-2, 19-4 Bui-Thanh Tan, 17-5 Awanou Gerard, 29-2 B¨urgerRaimund, 12-2 Ayuso de Dios Blanca, 7-2 Burman Erik, 11-7 Burstedde Carsten, 20-3, 20-11 Bachmayr Markus, 21-2 Badia Santiago, 16-2 Calhoun Donna, 20-3 Bai Yun, 10-2 Calo Victor, 15-7 Baiges Joan, 11-4 Cangiani Andrea, 18-2 Ballani Jonas, 21-3 Carey Varis, 10-3 Bandara Kosala, 4-17 Carraro Thomas, 22-3, 30-5 Banjai Lehel, 35-2, 35-3 Carstensen Carsten, 2-7, 9-2, 9-3, 17-6 Bank Randolph E., 17-19 Carter Edwards H., 33-6 Banks H. Thomas, 22-2, 22-4 Case Elizabeth, 35-8 B¨ansch Eberhard, 2-6 Cavalli Fausto, 26-4 Banz Lothar, 28-14 Cavallini Nicola, 2-11 Barreira Raquel, 31-4 Cec´ılioDiogo Lira, 2-8 Barrenechea Gabriel R., 11-2, 11-5 Cecot Witold, 17-24 Bauman Paul T., 10-3 Cerm´akMartin,ˇ 28-3 Bause Markus, 11-6 Cesmelioglu Aycil, 8-2 Beir˜aoda Veiga L., 18-2 Chalmers Noel, 34-4 Beir˜aoda Veiga L., 15-3, 18-4 Chandler-Wilde Simon, 2-12, 25-3, 25-4 Bercovier Michel, 15-5 Charrier Julia, 33-3 Beremlijski P., 28-2 Chatzipantelidis Panagiotis, 18-6 Berrone Stefano, 30-4 Chen Jingmin, 13-2 Berzins Martin, 20-2 Chen Ronald, 17-11 Bespalov Alex, 33-2 Chernov Alexey, 17-7, 28-4 Betcke T., 4-2 Childs. P.N., 2-12 Beuchler Sven, 3-5 Chkadua Otar, 5-2 Bierig Claudio, 28-4 Chrysafinos Konstantinos, 3-5

35-11 Chung Eric, 7-3 Feng Xiaobing, 7-5, 29-3 Ciarlet Patrick, 14-4 Fierro Francesca, 10-5 Cirak Fehmi, 4-17 Flaig Thomas G., 3-2, 3-3 Cockburn Bernardo, 8-2, 8-5, 34-6 Fonrasier Massimo, 2-5 Codina Ramon, 11-4, 16-2 Franz Sebastian, 11-9 Collier Nathan, 15-7 Frei Stefan, 3-7 Costabel Martin, 1-5 Friedmann Elfriede, 22-3 Croce Roberto, 28-11 Frutos Javier de, 11-9 Fu Zhixing, 12-3 Dahmen Wolfgang, 21-2 F¨uhrerThomas, 4-5, 23-6 Dalcin Lisandro, 15-7 Funke Simon W., 30-8 Dallmann Helene, 11-8 Dauge Monique, 9-4 Gahalaut Krishan, 15-9 Davies Penny J, 35-4 Gallistl Dietmar, 9-3 Davydov Oleg, 29-3 Gander Martin, 17-9, 25-6 Dawson Clint N, 1-1 Ganesan Sashikumaar, 11-10 Debussche Arnaud, 33-3 Ganesh M., 4-8 Deckelnick Klaus, 3-6, 31-2 Ganis Benjamin, 30-9 Dedner Andreas, 7-4, 31-3 Garc´ıa-Archilla Bosco, 11-9, 11-11 Demaret Laurent, 13-3 Garcia-Aznar Jose Manuel, 27-2 Demkowicz Leszek, 17-6, 17-8, 17-22 Gardini Francesca, 2-11 Demlow Alan, 16-2 Gastaldi Lucia, 2-11 Devloo Philippe R B, 2-8 Gedicke Joscha, 9-2, 9-3 Dickopf Thomas, 20-6 George Uduak, 27-4 Diening Lars, 23-4 Georgoulis Emmanuil H., 7-5, 23-7, 26-5 Di Rienzo Luca, 15-6 Gerds Peter, 9-6 Dolejˇs´ıV´ıt,2-9, 23-5 Gerecht Daniel, 22-3 Dolgov Sergey V., 21-4, 21-8 Geveler Markus, 20-4 Dost´alZdenˇek,28-5, 28-6, 28-13 Ghattas Omar, 17-5, 20-11 Drasdo Dirk, 27-7 Giani Stefano, 9-7, 9-8 Drouet Guillaume, 28-8 Gimperlein Heiko, 4-9 Duncan Dugald B, 2-10, 35-4 Gippert Sabrina, 24-2 Gl¨afke Matthias, 35-5 Egger Herbert, 17-7 Goldys Beniamin, 12-4 El-Kacimi A., 2-16 Goll Christian, 30-5 Elfverson Daniel, 18-7 Gomes Jefferson L.M.A., 30-2 Elliott Charles M., 1-6, 31-2, 31-4 Gomez-Benito Maria Jose, 27-2 Elsheikh Ahmed H., 30-6, 32-2 Gopalakrishnan Jay, 17-6, 17-8–17-10 Engstr¨omChristian, 9-5 Gorman Gerard J., 30-2 Engwer Christian, 30-7 Graham I. G., 25-6 Eriksson T., 17-27 Grasedyck Lars, 9-6, 21-3 Ern Alexandre, 10-4, 23-5, 33-4 Greenwald Steve E., 22-2, 22-4 Esterhazy Sofi, 25-5 Grohs Philipp, 13-4 Evans John A., 15-8 Groth Samuel, 25-7 Fahlke Jorrit, 20-4 Grubiˇsi´cLuka, 9-7, 9-8 Fankhauser Thomas, 14-5 Gruhne Volker, 35-3 Farrell Patrick E., 30-8 Grundel Sara, 13-2 Faustmann Markus, 4-4 Grzhibovskis Richards, 4-14, 5-3 Feischl Michael, 4-5, 4-7, 23-6 Gudi Thirupathi, 3-7, 7-6, 16-2

35-12 Gunzburger Max, 1-7 Kenz Zackary, 22-2–22-4 Guo Wei, 34-2 Khoromskaia Venera, 21-6 Guzm´anJohnny, 7-6, 16-2 Khoromskij Boris, 21-7 Gwinner Joachim, 28-9 Kirby Robert M., 34-3 Klawonn Axel, 24-2, 24-3 Hagstrom Thomas, 17-11 Kleiss S.K., 15-10 Hakula Harri, 9-7, 17-12 Kl¨ofkorn Robert, 7-4 Hannukainen Antti, 25-9 Knobloch Petr, 11-5 Harasim Petr, 2-2 K¨ohlerKaroline, 2-7 Harder Christopher, 18-8 Koppenol D.C., 27-3 Hardering Hanne, 13-4 Kopteva Natalia, 14-8 Hesthaven J. S., 4-8 Kornhuber Ralf, 28-10 Heuer Norbert, 4-9, 4-10, 12-3, 17-13 Kossioris G.T., 26-6 Hewett David, 25-3, 25-4, 25-7 Kostin Georgy, 19-3, 19-6 Hiemstra Ren´e,15-8 Kotas Petr, 28-11 Hild Patrick, 28-8 Kov´aˇrPetr, 4-11 Hinze Michael, 3-6 Kov´aˇrov´aTereza, 4-11 Hiptmair Ralf, 18-9 Kozubek Tom´aˇs,28-2, 28-5, 28-6, 28-13 Hoteit Ibrahim, 30-6 Krause Dorian, 20-6 Houston Paul, 17-14 Krause Rolf, 20-6, 28-11 Howarth C. J., 2-12 Krivodonova Lilia, 34-4 Hrkac Gino, 6-4 Kruse Carola, 22-2, 22-4 Hu Jonathan, 33-6 Kubatko Ethan, 8-3 Hu Shuhua, 22-2, 22-4 KuˇceraV´aclav, 2-14 Huber Martin, 25-10 Kuerten J.G.M., 8-6 Hughes Thomas J.R., 15-8 Kumar B. V. Rathish, 2-22 Hunsicker Eugenie, 14-6, 14-7 Kumar Dinesh, 17-16 Ippisch Olaf, 20-4 Kunoth Angela, 33-5 Isaac Tobin, 20-11 Kusner Robert, 13-2 Izs´akFerenc, 34-2 Kuzmin Dmitri, 11-15 Kyza Irene, 26-3 Jackson Matthew D., 30-2 Jagiella Nick, 27-7 Laghrouche O., 2-16 Janssen B¨arbel, 26-5 Lahaye D., 25-11 Jansson Johan, 2-6 Lakkis Omar, 23-7, 26-6, 29-4 Javierre Etelvina, 27-2 Laliena Antonio, 35-2 Jensen Max, 29-4 Langdon Stephen, 2-12, 25-3, 25-4, 25-7 Jeon Youngmok, 17-15 Lanser Martin, 24-2 Jimack Peter K., 20-5 Larsson Stig, 33-5 John Volker, 11-5, 11-11, 11-12 Le Kim-Ngan, 12-4 Juntunen Mika, 30-9 Ledger P.D., 17-17 J¨uttlerBert, 15-12, 15-15 Lee Chang-Ock, 2-15 Lehrenfeld Christoph, 17-23 Kamaludin Sarah, 2-3 Le MaˆıtreOlivier, 33-4 Kanschat Guido, 7-7, 20-6 Leykekhman Dmitriy, 3-8, 3-9, 16-3 Karakatsani Fotini, 2-6 Li Buyang, 26-7 Karkulik Michael, 4-5, 4-10 Li Hengguang, 14-6, 14-7, 16-4 Kavaliou Klim, 11-12 Li Jichun, 17-18 Kazeev Vladimir, 21-5 Li Xiaozhou, 34-5

35-13 Lian Haojie, 15-13 M¨ullerBenjamin, 24-3 Liertzer Matthias, 25-5 Muga Ignacio, 17-8, 17-10 Linke Alexander, 11-13 Mustapha Kassem, 34-6, 34-7 Lionheart W.R.B., 17-17 Lipnikov Konstantin, 8-4, 18-10 Naldi Giovanni, 26-4 Liu Xuefeng, 9-9 Nannen Lothar, 25-10 Lombardi Ariel, 14-3 Nappi Angela, 8-3 Lube Gert, 11-8, 11-17 Naß Martin, 3-10 Lubich Christian, 35-3 Nataraj Neela, 3-7 Luk´aˇsDalibor, 4-11 Natroshvili David, 5-6 Neckel Tobias, 20-8 Madhavan Pravin, 31-3 Neethling Stephen J., 30-2 Madzvamuse Anotida, 27-4, 31-4 Neic Aurel, 20-9 Mahmood M., 2-16 Neilan Michael, 12-5, 16-5, 29-5 Maischak Matthias, 28-12, 35-5 Neitzel Ira, 3-10 Makridakis Charalambos, 2-6, 23-7, 26-2 Neum¨ullerMartin, 17-9 M˚alqvistAxel, 18-7 Nguyen Hieu, 17-19 Mal´yLuk´aˇs,4-11 Nguyen Ngoc Cuong, 8-2 Manni Carla, 15-11 Nicaise Serge, 3-2, 14-9 Mantzaflaris Angelos, 15-12 Niemeyer Julia, 2-19 Manzini Gianmarco, 18-2–18-4, 18-10 Niemi Antti H., 17-20 Marini L. Donatella, 18-5 Nigam Nilima, 17-20 Marini L.D., 18-2 Nistor Victor, 14-6, 14-7 Markopoulos Alexandros, 28-2, 28-6 Nochetto Ricardo H., 10-6, 26-3 M´arquezAntonio, 12-5 Novo Julia, 11-9, 11-11 Mateos Mariano, 3-4 Matthies Gunar, 2-4, 11-14 Oden J. Tinsley, 1-9 Mazzieri Ilario, 7-2 Of G¨unther, 4-13, 4-17 Meddahi Salim, 9-10, 12-5, 17-13 Olivares Nicole, 17-10 Medvedeva Tatyana, 8-5 Ostien Jakob T., 33-6 Mehrmann V., 9-3 Ovall Jeffrey S, 9-7, 9-8, 16-4 Mehta Dwij, 22-2 Oyarz´uaRicardo, 12-6 Meidner Dominik, 3-9 Page Marcus, 6-4, 6-5 Meir Ammon J., 31-4 Pain Christopher C., 30-2 Melenk Jens Markus, 4-4, 4-12, 7-7, 17-8, Paredes Diego, 18-8 17-19, 25-5 Pardo David, 15-7, 17-8 Merta Michal, 4-11, 28-3 Park Eun-Hee, 2-15 Messner Michael, 35-6 Park Eun-Jae, 17-15 Micheletti Stefano, 2-5 Parsania Asieh, 7-7 Miedlar Agnieszka, 9-3, 9-8 Pasciak Joseph E., 25-2 Mikhailov Sergey E., 5-3, 5-4 Paszynski Maciej, 15-7 M¨ollerMatthias, 2-18 Patra A. K., 17-16 Moiola Andrea, 18-9 Payne Mark, 2-10 Monk Peter, 1-8 Pelosi Francesca, 15-11 Mora David, 9-10 Pencheva Gergina, 30-9, 30-11 Moser Robert, 17-22 Peraire Jaime, 8-2 Mostaghimi Peyman, 30-2 Percival James R., 30-2 M¨ullerBenjamin, 24-5 Perotto Simona, 2-5 Mueller Eike, 20-7 Perugia Ilaria, 18-9, 26-4

35-14 Petit Julien, 17-20 Sander Oliver, 24-4, 28-10 Petras Argyrios, 17-20 Sangalli Giancarlo, 1-4, 15-3, 17-4 Pfefferer Johannes, 3-4, 3-10 Sangwan Vivek, 2-22 Phillips Joel, 17-20 Santos Erick Raggio Slis, 2-8 Phipps Eric, 33-6 Saurin Vasily, 19-3, 19-6 Pieper Konstantin, 3-11 Sauter Stefan A., 7-7, 9-10, 35-7 Pieraccini Sandra, 30-4 Savostyanov Dmitry V., 21-4, 21-8 Pitman E. B., 17-16 Sayas Francisco-Javier, 4-9, 12-3, 12-5, 35- Poletti Valentina, 28-11 2, 35-3 Portero Laura, 30-3 Schanz Martin, 35-6 Posp´ıˇsilLuk´aˇs,28-13 Schedensack Mira, 9-3 Potse Mark, 20-6 Scheichl Robert, 20-7 Powell Catherine, 33-2 Schieweck Friedhelm, 11-15 Praetorius Dirk, 4-4, 4-5, 4-7, 4-12, 6-5, Schillinger Dominik, 15-8 23-6 Schillings Claudia, 33-7 Pryer Tristan, 29-4 Schindele Dominik, 19-2 Schmidt Alfred, 10-5 Qiu Jianxian, 34-2 Schmidt Gunther, 4-3 Qiu Jing-Mei, 34-2 Schneider Ren´e,2-23 Qiu Weifeng, 8-2, 17-9 Sch¨oberl Joachim, 17-23, 20-10, 25-10 Quarteroni Alfio, 7-2 Sch¨otzauDominik, 12-6 Rachowicz Waldemar, 17-21, 17-27 Schr¨oderJ¨org,24-3, 24-5 Radtke Patrick, 24-3 Schr¨oderAndreas, 28-14 Raeini Ali Q, 30-10 Schugart Richard, 27-5 Raisch Alexander, 13-5 Schwab Christoph, 18-9, 33-7 Rankin Richard, 11-2, 23-2 Schwarz Alexander, 24-3, 24-5 Rannacher Rolf, 3-7 Schwegler Kristina, 11-6 Ranner Tom, 31-2 Schweiger M., 4-2 Rauh Andreas, 19-2, 19-4 Scial`oStefano, 30-4 Scott Michael A., 15-13 Repin Sergey, 23-8 ˇ Rhebergen Sander, 7-9, 8-5 Sebestov´aIvana, 2-9 Rheinbach Oliver, 24-2, 24-3 Segeth Karel, 2-23 Richter Thomas, 32-2, 32-3 Selgas Virginia, 2-21 Riviere Beatrice, 7-8 Sellier A., 4-15 Rjasanow Sergej, 4-14 Senkel Luise, 19-4 Roberts Nathan V., 17-22 Serafin Marta, 17-24 Rodr´ıguez Rodolfo, 9-10 Sharma Natasha, 7-7 R¨osch Arnd, 3-4, 3-10 Shaw Simon, 22-2, 22-4 Romito M., 26-6 Sheikh A. H., 25-11 Rotter Stefan, 25-5 Si Hang, 11-16 Ruede Ulrich, 1-10 Siebert Kunibert G., 20-10, 23-8 Ruiz-Baier Ricardo, 12-2 Silvester David, 33-2 Russo Alessandro, 18-2, 18-5 Simeon Bernd, 2-19, 15-15, 24-6 Ryan Jennifer K., 34-7 Simpson Robert N., 15-13 Singh Gurpreet, 30-11 Sadhanala Veeranjaneyulu, 3-7 Slodiˇcka Marian, 6-6 Saeed Abid, 29-3 Smears Iain, 29-4 Salgado A.J., 10-6 Smigaj´ W., 4-2 Salgado Pilar, 2-21 Sofo Jorge, 14-7

35-15 Sokolov Andriy, 31-6 van ’t Wout Elwin, 35-10 Speleers Hendrik, 15-11, 15-14 Vassilev Danail, 8-4 Spence E. A., 25-6 V´azquezRafael, 1-4, 15-3, 15-6, 17-4 Srivastava Shweta, 11-10 Veeser Andreas, 10-5, 16-6 Stadler Georg, 20-3, 20-11 Veit Alexander, 35-7 Stamm B., 4-8 Vejchodsk´yTom´aˇs,23-3 Starke Gerhard, 24-3, 24-5, 24-7 Verani Marco, 18-11 Steeger Karl, 24-3, 24-5 Verf¨urthR¨udiger,16-6 Steinbach Olaf, 3-11, 4-13, 4-17, 17-9 Vermolen Fred J., 27-3, 27-6 Stephan Ernst P., 12-6, 28-15 Vexler Boris, 3-8, 3-9, 3-11, 16-3 St´ephanouAngelique, 27-4 Vignon-Clementel Irene, 27-7 Stinner Bj¨orn,31-3 Vlach Oldˇrich, 28-5, 28-6 Storath Martin, 13-3 Vohral´ıkMartin, 2-9, 10-4, 23-5 Styles Vanessa, 31-5 Vuik C., 25-11 Suess Dieter, 6-4 Vuik Kees, 35-10 Sun Weiwei, 26-7 Vuong Anh-Vu, 15-15 Sundar Hari, 20-11 Sung Li-yeng, 3-12 Wacker Benjamin, 11-17 Svyatskiy Daniil, 18-10 Walkley Mark A., 20-5 Wang Hongrui, 17-25 Takacs Thomas, 15-15 Wang Jilu, 17-25 Tantardini Francesca, 16-6 Warburton T., 17-26 Taus Matthias, 15-13 Weeger Oliver, 24-6 Tausch Johannes, 35-6, 35-8 Weile Daniel S., 35-9 Teckentrup Aretha, 33-7 Weinmann Andreas, 13-3 Thomas Derek C., 15-13 Weiss Wolfgang, 35-6 Tian Lulu, 8-6 Wheeler Mary F., 30-6, 30-9, 30-11, 32-2 Tobiska Lutz, 11-12, 11-14 Whiteman John R., 22-2, 22-4 Tollit Brendan S., 30-2 Wick Thomas, 32-2, 32-3 Tomar Satyendra, 15-9, 15-10, 20-11 Wihler Thomas P., 7-10, 14-5, 26-5 Torres H´ector,12-2 Wilcox Lucas C., 20-3 Tran Thanh, 12-4 Winkler Max, 14-3 Trevelyan J., 2-16 Winther Ragnar, 1-10 Tryoen Julie, 33-4 Wirz Marcel, 7-10, 14-5 Tuncer Necibe, 31-4 Wohlmuth B., 4-12 Turek Stefan, 31-6 Wollner Winnifried, 3-7, 32-3 Twigger Ashley, 25-3, 25-4 Wu Haijun, 14-4

Unger Gerhard, 4-16 Xenophontos Christos, 14-9, 17-19 Uski Vile, 14-6, 14-7 Xu Yan, 8-6

Vainikko Eero, 20-7 Ye Qiang, 9-11 Valdman Jan, 2-2 Yotov Ivan, 8-4, 30-3, 30-9, 30-12 Valentin Fr´ed´eric,18-8 Youett Jonathan, 28-10 Valero Clara, 27-2 Yu Thomas, 13-2 Van Bockstal Karel, 6-6 van der Heul Duncan R., 35-10 Zapletal Jan, 4-17 van der Vegt Jaap, 7-9, 8-5, 8-6 Zdunek Adam, 17-21, 17-27 van der Ven Harmen, 35-10 Zhang Jianfei, 20-5 van der Zee K.G., 10-6 Zigerelli Andrew, 13-2

35-16 Zlotnik Alexander, 17-2 Zou Jun, 14-4

35-17