Austrian Numerical Analysis Day

4-5 May 2017 Salzburg

Department of Mathematics University of Salzburg

4-5 May 2017 Salzburg Editors Lothar Banz Raoul Kutil Andreas Schr¨oder

Organization Lothar Banz [email protected]

Andreas Schr¨oder [email protected]

Department of Mathematics Paris Lodron University of Salzburg Hellbrunnerstr. 34 5020 Salzburg, Austria

Location Paris Lodron University of Salzburg Department of Mathematics Faculty of Natural Science (Gr¨unerH¨orsaal- HS403) Hellbrunnerstr. 34 5020 Salzburg, Austria

Supporting Organizations

Paris Lodron University of Salzburg

Department of Mathematics Preface

Following in the spirit and the tradition of previous events in this series of workshops, the 13th Austrian Numerical Analysis Day will be organized by and will take place at the Department of Mathematics of the Paris Lodron University of Salzburg on 4th and 5th of May 2017.

The goal of this workshop is to inform about research activities in the ﬁelds of numerical analysis and applied mathematics. Scientists from Austrian universities and other research institutions in particular are invited to present their research and discuss their ideas. Apart from strengthening already well established contacts this annual workshop should also provide an opportunity to start new collaborations.

Lothar Banz and Andreas Schr¨oder Salzburg, May 2017

Contents

Internet Access 7 Conference Dinner8 Program 10 Abstracts 12 Benjamin Stadlbauer: Brownian-dynamics simulations of nonopore protein sensing 12 Gregor Mitscha-Baude: Simulation of nanopores with the Poisson-Nernst-Plack- Stokes and adaptive finite elements ...... 13 Stefan Rigger: Approximation of Multisymmetric Functions ...... 14 Gudmund Pammer: Computing Cubature Formulas for Multisymmetric Functions and Applications to Stochastic Partial Differential Equations ...... 15 Amirreza Khodadadian: Optimal multi-level Monte Carlo method for the stochastic drift-diffusion-Poisson system ...... 16 Boaz Blankrot: Multiple scattering approach for dielectric metamaterial analysis . . 17 Lukas Kogler: ASC-AMG, a parallel AMG-solver for Netgen/NGSolve ...... 18 Bernd Schwarzenbacher: Algebraic Multigrid for Maxwell’s Equations ...... 19 Daniel Jodlbauer: Robust and Efficient Solvers for Fluid-Structure Interaction . . . 20 Markus Gasteiger: ADI type preconditioners for the steady state inhomogeneous Vlasov equation ...... 21 Gerhard Kitzler: A tensor product framework for kinetic equations ...... 22 Darian M. Onchis: Numerical considerations of consistency and stability in spline- type spaces ...... 23 Christian Gerhards: Modeling and Numerical Aspects of Inverse Problems in Geo- magnetism ...... 24 Markus Schöbinger: Simulation of Eddy Currents in an Iron Ring Core Using a Multi-Scale Method ...... 25 Mario Luiz Previatti de Souza: Convergence result for IRGNM type method under a tangential cone condition in Banach space ...... 26 Paolo Di Stolfo: Dual weighted residual error estimation for the finite cell method . 27 Bernhard Endtmayer: Adaptive Mesh Refinement for Multiple Goal Functionals . 28 Joscha Gedicke: Residual-based a posteriori error analysis for symmetric mixed Arnold-Winther FEM ...... 29 Thomas Führer: On the DPG method for Signorini problems ...... 30 Nina Ovcharova: Numerical methods for nonmonotone contact problems in continuum mechanics ...... 31 Markus Faustmann: Local convergence of the boundary element method on polyhedral domains ...... 32

5 Thomas Apel: Superconvergent graded meshes: First results ...... 33 Philip Lukas Lederer: Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations ...... 34 Olaf Steinbach: Regularization error estimates for distributed control problems . . . 35 Clemens Hofreither: A Black-Box Algorithm for Fast Matrix Assembly in Isogeo- metric Analysis ...... 36 Linus Wunderlich: Isogeometric mortar methods in solid mechanics ...... 37 Svetlana Matculevich: Functional a posteriori error estimates and adaptivity for IgA schemes ...... 38 Ioannis Toulopoulos: Time Discontinuous Galerkin Multipatch Isogeometric Anal- ysis of Parabolic Diffusion Problems ...... 39 Stefan Dohr: Space-time boundary element methods for the heat equation ...... 40 Marco Zank: Space-Time Boundary Element Method for the Wave Equation . . . . 41 Bernhard Stiftner: Linear second order implicit-explicit time-integration of the (eddy-currents-)Landau-Lifshitz-Gilbert equation ...... 42 Harald Hofstätter: Time propagators for Schrödinger-type equations with expensive- to-evaluate nonlinear part ...... 43 List of participants 45

6 Internet Access

During the Austrian Numerical Analysis Day there will be free internet access in the lecture room during the seminar. Please enter the following in the login mask:

SSID: Plus Event User: AUNUDA Password: yalana98!

7 Conference Dinner

For the conference dinner, we have reserved a table at a nearby restaurant Sternbr¨authat provides a three course meal. The joint dinner at the traditional Austrian restaurant is an optional opportunity to discuss various topics.

Please note that the coupon included in the workshop documents covers the dinner and the ﬁrst drink. The menu is:

Strong beef broth with pancake stripes

Roast beef in onion gravy with spaetzle and green beans or Roasted brook trout ﬁlet on creamy kohlrabi with bulgur and braised tomato

Sweet cheese dumplings with sour cherry sauce

Dinner: Sternbr¨au– May 4, 2017 starting at 7:30pm Griesgasse 23, 5020 Salzburg. http://www.sternbrau.com/en/

8 How to get to Sternbr¨au:

It will take you approx. 6 minutes to get from the building where the conference is taking place to the bus stop ”Faistauergasse” on the main road (Alpenstraße). There you take the bus line 8 or 3 on the opposite side of the road. If you take the bus line 8, get off at the 6th bus stop ”Zentrum-Ferdinand Hanusch Platz”, otherwise get off at ”Theatergasse”. At ”Theater- gasse” you first have to cross the river via the Staatsbrücke and then turn right to get to the bus stop ”Zentrum-Ferdinand Hanusch Platz”, which is a two minute walk away from the Sternbräu.

Alternatively, it will take you about 30 minutes to walk. Just get to the main road, turn left (west) and follow the road until you have reached the bus stop ”Zentrum-Ferdinand Hanusch Platz”.

9 Program

Thursday (May 4th, 2017)

12:00-13:00 Registration 13:00-13:10 Opening 13:10-13:30 Benjamin Stadlbauer Brownian-dynamics simulations of nonopore protein sensing 13:30-13:50 Gregor Mitscha-Baude Simulation of nanopores with the Poisson-Nernst-Plack-Stokes and adaptive finite elements 13:50-14:10 Stefan Rigger Approximation of Multisymmetric Functions 14:10-14:30 Gudmund Pammer Computing Cubature Formulas for Multisymmetric Functions and Applica- tions to Stochastic Partial Differential Equations 14:30-14:50 Amirreza Khodadadian Optimal multi-level Monte Carlo method for the stochastic drift-diffusion- Poisson system 14:50-15:20 Coffee Break 15:20-15:40 Boaz Blankrot Multiple scattering approach for dielectric metamaterial analysis 15:40-16:00 Lukas Kogler ASC-AMG, a parallel AMG-solver for Netgen/NGSolve 16:00-16:20 Bernd Schwarzenbacher Algebraic Multigrid for Maxwell’s Equations 16:20-16:40 Daniel Jodlbauer Robust and Efficient Solvers for Fluid-Structure Interaction 16:40-17:00 Markus Gasteiger ADI type preconditioners for the steady state inhomogeneous Vlasov equation 17:00-17:15 Break 17:15-17:35 Gerhard Kitzler A tensor product framework for kinetic equations 17:35-17:55 Darian M. Onchis Numerical considerations of consistency and stability in spline-type spaces 17:55-18:15 Christian Gerhards Modeling and Numerical Aspects of Inverse Poblems in Geomagnetism 18:15-18:35 Markus Schöbinger Simulation of Eddy Currents in an Iron Ring Core Using a Multi-Scale Method 19:30 Joint Dinner at Sternbräu

10 Friday (May 5th, 2017)

08:30-08:50 Mario Luiz Previatti de Souza Convergence result for IRGNM type method under a tangential cone condition in Banach space 08:50-09:10 Paolo Di Stolfo Dual weighted residual error estimation for the finite cell method 09:10-09:30 Bernhard Endtmayer Adaptive Mesh Refinement for Multiple Goal Functionals 09:30-09:50 Joscha Gedicke Residual-based a posteriori error analysis for symmetric mixed Arnold- Winther FEM 09:50-10:10 Thomas Führer On the DPG method for Signorini problems 10:10-10:40 Coffee Break 10:40-11:00 Nina Ovcharova Numerical methods for nonmonotone contact problems in continuum mechanics 11:00-11:20 Markus Faustmann Local convergence of the boundary element method on polyhedral domains 11:20-11:40 Thomas Apel Superconvergent graded meshes: First results 11:40-12:00 Philip Lukas Lederer Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations 12:00-12:20 Olaf Steinbach Regularization error estimates for distributed control problems 12:20-13:20 Lunch Break 13:20-13:40 Clemens Hofreither A Black-Box Algorithm for Fast Matrix Assembly in Isogeometric Analysis 13:40-14:00 Linus Wunderlich Isogeometric mortar methods in solid mechanics 14:00-14:20 Svetlana Matculevich Functional a posteriori error estimates and adaptivity for IgA schemes 14:20-14:40 Ioannis Toulopoulos Time Discontinuous Galerkin Multipatch Isogeometric Analysis of Parabolic Diffusion Problems 14:40-15:10 Coffee Break 15:10-15:30 Stefan Dohr Space-time boundary element methods for the heat equation 15:30-15:50 Marco Zank Space-Time Boundary Element Method for the Wave Equation 15:50-16:10 Bernhard Stiftner Linear second order implicit-explicit time-integration of the (eddy-currents-) Landau-Lifshitz-Gilbert equation 16:10-16:30 Harald Hofstätter Time propagators for Schrödinger-type equations with expensive-to-evaluate nonlinear part 16:30 Closing

11 Brownian-dynamics simulations of nonopore protein sensing

Benjamin Stadlbauer TU Wien, Wiedner Hauptstraße 8, 1040 Wien, [email protected]

Gregor Mitscha-Baude TU Wien, Wiedner Hauptstraße 8, 1040 Wien, [email protected]

Clemens Heitzinger TU Wien, Wiedner Hauptstraße 8, 1040 Wien, [email protected]

Keywords: nanopores, Langevin simulations, Brownian dynamic simulations, stochastic diﬀerential equation

ABSTRACT

Nanopores are tiny holes in insulating membranes which connect two electrolyte chambers on each side. If an electric potiential is applied, it induces ion transport through this opening, which is measured. If a target molecule moves inside the nanopore, it partially blocks the channel and causes a reduction in the current trace. In this way, one can detect molecules such as DNA strands or proteins, where the amplitude, duration, and the shape of the event signal distinguishes between different types of particles. While nanopore DNA sensing is better understood, protein sensing still needs to be studied better. This also implies challenges for modeling and numerical analysis. To simulate protein sensing with nanopores, approaches such as molecular dynamics (MD) and simple Langevin simulations have been used. However, MD simulations are computationally extremely expensive and therefore the physics are sometimes simplified in order to reduce runtime. Furthermore, simple Langevin simulations do not always produce realistic results because of the lack of important physical and chemical details. We present another model for modeling protein sensing, which is also based on the Langevin equation, a stochastic differential equation. In contrast, we also consider the contribution of the electroosmotic flow, which is computed with a continuum model [1]. In addition, we take non-constant anisotropic diffusivity of the ions as well as of the proteins into account, and therefore the results we obtain are much more realistic. In contrast to MD simulations, we can consider long-time unspecific binding of the protein inside the nanopore. Additionally, because our simulations are much faster, it is possible to obtain hundreds or thousands of events and compile statistics, which is necessary, since the movement of a target molecule is a stochastic process. Hence it has become possible to simulate the trajectories of the particles and therefore the time-dependent event signals in various setups and furthermore to examine the distributions of the dwell times of the events and also of the amplitudes of the current reductions. With our model we are not only able to reproduce experimental data, but also to answer open questions regarding the interpretation of certain events which have been observed in very recent experiments.

REFERENCES

[1] Gregor Mitscha-Baude, Andreas Buttinger-Kreuzhuber, Gerhard Tulzer, and Clemens Heitzinger. Adaptive and iterative methods for simulations of nanopores with the PNP- Stokes equations. Journal of Computational Physics, 338:452-476, 2017.

12 Simulation of nanopores with the Poisson-Nernst-Plack-Stokes and adaptive ﬁnite elements

Gregor Mitscha-Baude TU Wien, Wiedner Hauptstraße 8–10, [email protected]

Benjamin Stadlbauer TU Wien, Wiedner Hauptstraße 8–10

Clemens Heitzinger TU Wien, Wiedner Hauptstraße 8–10

Keywords: nanopores, PDE models for transport, goal-oriented adaptivity

ABSTRACT

Nanopores are tiny holes which enable ions and biological molecules to flow through an otherwise insulating membrane. They can be used for detection of large molecules by monitoring the electrical current through the pore, because these molecules block ion flow when they enter the nanopore and cause a measurable drop in current. Discriminating molecules by their current signatures has made it possible to sequence DNA. The sequencing of proteins with nanopores is still an open problem, and would have huge scientific and medical implications. The phenomena associated with nanopore sensing offer a host of fascinating problems for modeling, analysis and scientific computing. We employ the Poisson and Nernst-Planck equations to model ion current, the Stokes system to describe the flow of water around particles, and a Fokker-Planck equation to model the stochastic transport of proteins through a nanopore. All these equations are coupled to each other, including nonlinear interactions in the PoissonNernst-Planck-Stokes (PNPS) system. First, we present our finite element approach to solve the PNPS equations on realistic 3D nanopore geometries [1]. Since the focus is on obtaining quantities of practical interest, namely the ion current and electrophoretic force induced on a protein, we develop a goal-oriented adaptive mesh refinement strategy. Second, we investigate non-constant diffusion constants of ions and proteins. Diffusion constants enter both the PNPS and the Fokker-Planck equations. We find that they are reduced considerably inside a nanopore. To compute them, we again need to solve a Stokes equation for many locations of the ion/protein. In this way, we obtain a stack of three PDE models on top of each other, where a large number of solutions of one PDE has to be evaluated to determine the coefficients of the next one: The Stokes equation determines the diffusion constant in the PNPS system, which in turn determines the force on a protein in the Fokker-Planck model. This poses the not entirely trivial side question of how to interpolate a coefficient from as few evaluations as possible in an arbitrary 3D geometry.

REFERENCES

[1] Gregor Mitscha-Baude, Andreas Buttinger-Kreuzhuber, Gerhard Tulzer, and Clemens Heitzinger. Adaptive and iterative methods for simulations of nanopores with the PNPStokes equations. Journal of Computational Physics, 338:452-476, 2017.

13 Approximation of Multisymmetric Functions

Stefan Rigger TU Wien, A-1040 Vienna, Austria, [email protected]

Gudmund Pammer TU Wien, A-1040 Vienna, Austria, [email protected]

Clemens Heitzinger TU Wien, A-1040 Vienna, Austria, [email protected]

Keywords: quadrature and cubature formulas, multivariate integration, permutation-invariance, multisymmetric polynomials

ABSTRACT

Many interesting applications in physics exhibit symmetries that can be exploited to reduce computational eﬀort. Therefore, we consider a mathematical setting that reﬂects these symmetry properties. We introduce the following notion of invariance under permutations: for a subgroup G of SN , we call an N-variate real-valued function G-invariant if f ◦ σ = f for every σ in G. We prove that the space of G-invariant continuous functions is a Banach space that contains the space of G-invariant polynomials as a dense subspace. An analogous theorem can be shown in the case of spaces of p-integrable functions. We also show that the Taylor polynomials of a G-invariant function centered at a G-invariant point are G-invariant. These results naturally motivate the idea of cubature rules for G-invariant functions, where we demand that a rule of order d should integrate every G-invariant polynomial of degree less than or equal to d exactly. We prove error bounds for cubature formulas of this type. Furthermore, we investigate an important special case of G-invariance, namely multisymmetry groups. Finally, we prove that in a certain sense, there is no curse of dimensionality on spaces of multisymmetric polynomials (in contrast to ordinary polynomial spaces).

REFERENCES

14 Computing Cubature Formulas for Multisymmetric Functions and Applications to Stochastic Partial Diﬀerential Equations

Gudmund Pammer TU Wien, A-1040 Vienna, Austria, [email protected]

Stefan Rigger TU Wien, A-1040 Vienna, Austria, [email protected]

Clemens Heitzinger TU Wien, A-1040 Vienna, Austria, [email protected]

Keywords: quadrature and cubature formulas, multivariate integration, permutation-invariance, multisymmetric polynomials

ABSTRACT

Many interesting applications in physics that can be modeled by stochastic partial differential equations exhibit symmetries that can be exploited to reduce computational effort. Especially in the case of stochastic PDE, the numerical solution of such problems requires the numerical integration of high-dimensional integrals, since the Curse of Dimensionality is encountered frequently and problems turn out to be computationally highly demanding. By exploiting permutation- invariance properties, the complexity of the integration problem can be significantly reduced, and problems that are unsolvable using a naive approach become tractable as in [3]. We demonstrate how cubature formulas for multisymmetric functions can be calculated in prac- tice, making use of the special structure of multisymmetry groups. We compare cubature formulas for multisymmetric functions to tensor product rules in the low-dimensional case and to Monte-Carlo and sparse-grid methods in the high-dimensional case. Finally, we compute the expectation of the solution of a stochastic partial differential equation both using a quasiMonte- Carlo method and our proposed formulas, discussing the benefits of our method. In the case of smooth multisymmetric functions, the results indicate that the proposed cubature formulas are highly efficient.

REFERENCES

[1] C. Heitzinger, G. Pammer and S. Rigger. Cubature formulas for multisymmetric functions and applications to stochastic partial diﬀerential equations. Submitted, 2017. [2] C. Heitzinger, G. Pammer and S. Rigger. Numerical solution of multisymmetric stochastic partial diﬀerential equations. In preparation, 2017. [3] D. Nuyens, G. Suryanarayana and M. Weimar. Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions. Advances in Computational Mathematics, 2016.

15 Optimal multi-level Monte Carlo method for the stochastic drift- diﬀusion-Poisson system

Amirreza Khodadadian TU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, [email protected]

Leila Taghizadeh TU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, [email protected]

Clemens Heitzinger TU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, [email protected]

Keywords: Silicon nanowire sensors, multi-level Monte Carlo, stochastic drift-diﬀusion-Poisson system

ABSTRACT

The stochastic drift-diffusion-Poisson system [1] serves as a leading example for the development of optimal numerical algorithms for systems of stochastic PDE. The model consists of the Poisson-Boltzmann equation to model the electrolyte and the drift-diffusion-Poisson system to model the charge transport in the transducer. Also, a reaction equation is applied to model the association of target molecules to the sensor surface and their dissociation. The optimal multi-level Monte Carlo (MLMC) is developed to obtain an accurate estimation of the expected value of the solution of the system [1]. This allows to find the optimal choice of discretization parameters. In other words, we minimized the overall computational cost for a prescribed total error i.e., spatial error (finite-element discretization) as well as statistical error (i.e., randomness of the system). Here, we define a global optimization problem which minimizes the computational complexity such that the error bound is less or equal to a given tolerance level. To further improve the computational efficiency, a randomized low-discrepancy sequence such as a randomly shifted lattice are applied as well [2]. The applications considered here are noise and fluctuations in silicon nanowire sensors and multi-gate transistors. The multi-level approach shows noticeable advantages compared to the single-level method, where for lower error bounds, the computational work is reduced by four orders of magnitude. The speed-up becomes better as the error tolerance decreases.

REFERENCES

[2] Leila Taghizadeh, Amirreza Khodadadian, and Clemens Heitzinger. The optimal multi- level Monte-Carlo approximation of the stochastic drift-diﬀusion-Poisson system. Com- puter Methods in Applied Mechanics and Engineering (CMAME), 318 (2017): 739-761. [1] Leila Taghizadeh, Amirreza Khodadadian, and Clemens Heitzinger. Optimal multi-level randomized quasi Monte-Carlo method for the stochastic drift-diﬀusion-Poisson system. pages 1–21. In preparation.

16 Multiple scattering approach for dielectric metamaterial analysis

Boaz Blankrot Institute for Analysis and Scientiﬁc Computing, TU Wien; Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria; [email protected]

Clemens Heitzinger Institute for Analysis and Scientiﬁc Computing, TU Wien; Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria; [email protected]

Keywords: Computational Electromagnetics, Integral Equation Methods, Julia

ABSTRACT

In this talk we consider electromagnetic scattering from a collection of many similar arbitrarily- shaped inclusions in the metamaterial regime (R/λ ≈ 0.5). We use Julia, a young high- performance programming language for numerical computing, to implement a solver for this problem in two dimensions. Generally, an integral-equation approach is appropriate for such open scattering problems, however this yields dense system matrices that are costly to solve. Additionally, moving or rotating the inclusions would require re-computation of a large portion of the system matrix. To address these issues, we combine the integral equation method with a multiple-scattering formulation [1], which replaces each inclusion with a scattering matrix that translates incoming multipole expansions to an outgoing one. In this approach the integral equation is solved locally in the preprocessing stage, and only once for each type of inclusion. For smooth inclusions in our regime the multipole expansion length is substantially smaller than the number of points used to discretize the boundary. Therefore, the number of degrees of freedom in the global system is reduced by an order of magnitude. We accelerate our solver by means of a fast multipole method, resulting in time complexity that scales roughly linearly with the number of inclusions. We describe how this tool can be used in the design and analysis of two-dimensional dielectric metamaterials. For example, we examine the Luneburg lens and its realization by means of a graded index photonic crystal [2]. In addition, the possibility of automating the design process by applying optimization techniques to this solver is discussed.

REFERENCES

[1] J. Lai, M. Kobayashi, and L. Greengard. A fast solver for multi-particle scattering in a layered medium. Optics Express, Vol. 22, 20481–20499, 2014. [2] F. Gauﬁllet and E.´ Akmansoyd. Graded Photonic Crystals for Luneburg Lens. IEEE Photonics Journal, Vol. 8, 1–11, 2016.

17 ASC-AMG, a parallel AMG-solver for Netgen/NGSolve

Lukas Kogler Vienna University of Technology, Vienna, [email protected]

Joachim Sch¨oberl Vienna University of Technology, Vienna, [email protected]

Keywords: AMG, MPI, FEM

ABSTRACT

We introduce ASC-AMG, which stands for ”Alternative Strong Connections AMG”, a MPI- parallel algebraic multigrid solver for Netgen/NGSolve which is currently under development. It can be seen as a variant of AMG by agglomeration where we take a new approach to define strong connections between degrees of freedom. In most traditional AMG solvers, the strength of connection between degrees of freedoms i and j is measured by the corresponding off-diagonal entry in the system matrix A. Our approach is more closely related to the underlying FEM-discretization. On the element matrix level, we compute equivalent but simpler structured replacement element matrices which, when assem- bled, give us a replacement matrix Aˆ for the entire FEM-System such that ||u||2 ≈ ||u||2 = A Aˆ P 2 ij wij(ui − uj) , where wij are the weights we use to measure strength of connection. We present our approach to parallel agglomeration based on the replacement matrix and show the formalism we use to describe the parallel nature of degrees of freedom in an MPI-based setting. We will also briefly show our approach to coarse grid interpolation, which also uses the replacement- matrix and allows for communication-less transfer between grid levels. Finally, we will show scalability results on clusters up to 2000 cores.

18 Algebraic Multigrid for Maxwell’s Equations

Joachim Sch¨oberl TU Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, [email protected]

Bernd Schwarzenbacher TU Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, [email protected]

Keywords: Maxwell’s equations, algebraic multigrid, ﬁnite element method

ABSTRACT

We present an algebraic multigrid method to solve large sparse systems of linear equations arising from Nédélecfinite element discretization of problems posed in H(curl, Ω) [1]. The main part is a new criterion, motivated by additive Schwarz theory, for a sensible mesh collapse algorithm to construct a hierarchy of coarse representations from the original fine mesh. The criterions aim is to preserve the homology of the domain. Together with a prolongation operator based on the paper of Reitzinger-Schöberl [2], which maps fine curl-free functions to coarse curl-free functions, we achieve, that the De Rham complex remains a complete sequence on each level. This for one gives us robustness in parameter jumps. Further we can now use a Hiptmair smoother [3] to obtain robustness in small regularization parameters. The method was implemented within the finite element software NGSolve (ngsolve.org). In the end we show numerical results for magnetostatic examples in 3D demonstrating parallel scalability on shared memory computers.

REFERENCES

[1] Peter Monk. Finite Element Methods for Maxwell’s Equations, Oxford University Press Inc. New York, 2003 [2] Reitzinger, S. and Sch¨oberl, J. An algebraic multigrid method for ﬁnite element discretizations with edge elements. Numerical Linear Algebra with Applications, Vol. 9, 223–238, 2002. [3] Ralf Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal., Vol. 36 204–225, 1999.

19 Robust and Eﬃcient Solvers for Fluid-Structure Interaction

Daniel Jodlbauer Johannes Kepler University Linz, Doctoral Program “Computational Mathematics”, Altenberger Straße 69, 4040 Linz, [email protected]

Ulrich Langer Institute of Computational Mathematics JKU Linz, Altenbergerstraße 69, 4040 Linz, [email protected]

T. Wick Centre de Math´ematiquesAppliqu´ees(CMAP), Ecole Polytechnique, Route de Saclay, 91128 PALAISEAU Cedex, France, [email protected]

Keywords: solver, monolithic, ﬂuid-structure-interaction, ALE, parallelization

ABSTRACT

Fluid-structure-interaction problems have a wide range of applications, but their efficient solution remains challenging. In this work we provide all details necessary for a monolithic ALE implementation using the finite element library deal.II. To actually solve the arising linear systems, we develop a preconditioner based on an approximate block-wise LU-factorization, splitting the coupled system of equations into its natural fluid, solid and mesh sub-problems. Nu- merical results illustrate the robust convergence with respect to different material parameters and mesh-size h, and with an acceptable dependence on the time-step size ∆t. Furthermore, some preliminary results regarding parallelization are shown.

REFERENCES

[1] D. Jodlbauer, T. Wick. Monolithic FSI. Radon Series on Comp. App. Math, Vol. 20, 2017. [2] D. Jodlbauer. Robust Preconditioners for Fluid-Structure Interaction. Master thesis, 2016 [3] U. Langer and H. Yang. Robust and eﬃcient monolithic ﬂuid-structure-interaction solvers. Int. J. Numer. Meth. Engng., Vol. 108, 303–325, 2016. [4] T. Wick. Solving Monolithic Fluid-Structure Interaction Problems in Arbitrary La- grangian Eulerian Coordinates with the deal.II Library. Archive of Numerical Software, Vol. 1, 1–19, 2013.

20 ADI type preconditioners for the steady state inhomogeneous Vlasov equation

Markus Gasteiger University of Innsbruck, Technikerstr. 13, A-6020 Innsbruck, [email protected] in collaboration with Lukas Einkemmer University of Innsbruck

Alexander Ostermann University of Innsbruck

David Tskhakaya Technical University of Vienna

Keywords: Vlasov equation, preconditioning, iterative methods, plasma physics.

ABSTRACT

The purpose of this work is to ﬁnd numerical solutions of the steady state inhomogeneous Vlasov equation. This problem has a wide range of applications in the kinetic simulation of non-thermal plasmas. However, the direct application of either time stepping schemes or iterative methods (e.g. Krylov based methods or relaxation schemes) is computationally expensive. In the former case the slowest timescale in the system forces us to perform a long time integration while in the latter case a large number of iterations is required. In this paper we propose a preconditioner based on an alternating direction implicit (ADI) type splitting method. This preconditioner is then used with both GMRES and Richardson iteration. The resulting numerical schemes scale almost ideally (i.e. the computational eﬀort is proportional to the number of grid points). Numerical simulations conducted show that this can result in a speedup of close to two orders of magnitude (even for intermediate grid sizes) with respect to the unpreconditioned case. In addition, we discuss the characteristics of these numerical methods and show the results for a number of numerical simulations.

REFERENCES

M. Gasteiger, L. Einkemmer, A. Ostermann, and D. Tskhakaya. Alternating direction implicit type preconditioners for the steady state inhomogeneous Vlasov equation. Journal of Plasma Physics, 83(1), 2017.

21 A tensor product framework for kinetic equations

Gerhard Kitzler Vienna UT, Wiedner Hauptstraße 8-10, 1030 Wien, [email protected]

Joachim Sch¨oberl Vienna UT, Wiedner Hauptstraße 8-10, 1030 Wien, [email protected]

Keywords:

ABSTRACT

In the talk we present a tensor product framework for the solution of higher dimensional problems. We show efficient evaluation of bilinear forms and solution fields and that both can be reduced to the individual components of the tensor product. For that reason there is no need to implement high dimensional finite elements, only the tensor product structure needs to be treated properly. The bilinear forms for continuous as well as discontinuous Galerkin methods can be specified within the Python interface NGS-Py of the finite element library NGSolve. Finally we demonstrate the usability of the framework by application to kinetic equations gov- erning carrier transport in semiconductors.

REFERENCES

[1] www.ngoslve.org.

22 Numerical considerations of consistency and stability in spline- type spaces

Darian M. Onchis University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, [email protected]

Simone Zappala University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, [email protected]

Keywords: spline-type systems, signal decomposition, numerical realizations, stability and consistency, Gabor system

ABSTRACT

Spline-type spaces are a class of shift-invariant spaces distinguished by the possession of a Riesz basis which consists of a set of translates along some lattice of a ﬁnite family of atoms. In this spaces, we characterize generating sets and subgroups that ensure the invertibility of the synthesis operator; we call this property consistency. Because the characterization of a spline type space belongs to its representation in the Fourier domain, we established a theoretical and numerical method to select proper modulations for an input function or signal. Motivated by this selection in the frequency domain that can lower the computational complexity of traditional frame decomposition such as Gabor frame, we continued the study of the stability of spline- type systems through the study of continuity in the Frobenius norm of the synthesis operator under deformation of the generating set in the Fourier domain. We tested numerically the results for the construction of approximate dual Gabor-like frames. We present the advantages of this approach in both ﬂexibility and speed. The method allows in a natural way to handle non standard Gabor constructions like non-uniformity in frequency and the reductions of the number of used modulations.

REFERENCES

[1] H. Feichtinger and D. Onchis. Constructive realization of dual systems for generators of multi-window spline-type spaces. Journal of computational and applied mathematics, Vol. 234.12, 3467–3479. 2010. [2] K. Gr¨ochenig. Foundations of time-frequency analysis, Birkh¨auserBasel, 2001. [3] A Ron and Z. Taylor. Generalized shift-invariant systems. Constructive approximation, Vol. 22.1, 1–45. 2005.

23 Modeling and Numerical Aspects of Inverse Poblems in Geomag- netism

Christian Gerhards University of Vienna, Computational Science Center, Oskar-Morgenstern Platz 1, 1090 Wien

Keywords: Inverse Potential Field Problems, Approximation on the Sphere, Geomagnetism

ABSTRACT

Several satellite missions have provided and are providing continuous measurements of the Earth’s magnetic field over the last 15-20 years. A major task is the extraction of the different contributions of the measured field (e.g., the contribution due to dynamo effects in the Earth’s core, the contribution stemming from magnetizations in the Earth’s crust, or the contribution produced by tidal ocean flow). After a brief introduction, this talk focuses on the separation of the core and the crustal contribution. That is, knowing the magnetic field B = Bcore + Bcrust on a sphere SR, is it possible to separate the contributions Bcore and Bcrust? In general, this problem is non-unique. We provide a modeling approach, based on the assumptions that Bcrust is generated by locally supported magnetizations on the spherical Earth’s surface, which yields uniqueness of the problem. Fur- thermore, we derive an optimization problem that leads to an approximation of Bcrust, given only the knowledge of B on SR. Eventually, we present first numerical examples based on radial basis function expansions.

24 Simulation of Eddy Currents in an Iron Ring Core Using a Multi- Scale Method

Markus Sch¨obinger Technische Universit¨atWien, Vienna, A-1040 Austria, [email protected]

Joachim Sch¨oberl Technische Universit¨atWien, Vienna, A-1040 Austria, [email protected]

Karl Hollaus Technische Universit¨atWien, Vienna, A-1040 Austria, [email protected]

Keywords: eddy current, multi-scale, nonlinear, network coupling

ABSTRACT

The nonlinear eddy current problem with network coupling - for a given voltage U ﬁnd the magnetic vector potential A(t) ∈ H(curl) and current I(t) ∈ R so that Z ∂ Z Z ν(A) curl A curl v dΩ + σAv dΩ = Kv dΓ Ω ∂t Ω Γ(Ω) Z ∂A IR + τ dΓ = U Γ(Ω) ∂t for every v ∈ H(curl) - is solved to simulate the eddy currents in an iron ring core. In order to reduce the number of degrees of freedom, cylindrical coordinates are used to model the radially symmetric domain using only two dimensions. Furthermore the single laminates of the iron core are not resolved in the mesh. Instead a multi-scale method is used to recover the local behavior. The quality of the simulation is checked using measurement data provided by the Institute of Electrical Machines of the RWTH Aachen.

REFERENCES

[1] A. Bensoussan, J. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Struc- tures. North-Holland, 2011. [2] K. Hollaus and J. Sch¨oberl. Homogenization of the eddy current problem in 2d. 14th Int. IGTE Symp., Graz, Austria, Sep. 2010, pp. 154–159.

25 Convergence result for IRGNM type method under a tangential cone condition in Banach space

Barbara Kaltenbacher Alpen-Adria Universitat Klagenfurt, 9020, [email protected]

Mario Luiz Previatti de Souza Alpen-Adria Universitat Klagenfurt, 9020, [email protected]

Keywords: Gauss Newton Method, Regularization, Discretization

ABSTRACT

This talk deals with a combined analysis of regularization and discretization of inverse problems in Banach spaces, specifically in the context of partial differential equations (PDEs). The relevant quantities - parameters and states - have to be discretized, e.g., by the finite element method, and the error due to this discretization has to be appropriately estimated and controlled by error estimators and mesh refinement. Thus one of the main challenges is here to take into account the interplay between mesh size, regularization parameter and data noise level. The focus on the PDEs setting is relevant to the adaptive discretization of the regularized problems. Hence, I will present convergence of the Iteratively Regularized Gauss Newton Method (IRGNM) in its classical Tikhonov version and in the IRGM Ivanov version under a tangential cone condition in Banach space setting. Convergence without source conditions has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or the spaces. I will also present how to obtain the a posteriori estimates and to achieve the prescribed accuracy by adaptive discretization using goal oriented error estimators. Such problems play a crucial role in numerous applications, ranging from medical imaging via nondestructive testing (e.g. electrical impedancy tomography) to geophysical prospecting (e.g. inverse water ground filtration), with the Banach space setting assigned by the inherent regularity of the sought coefficients as well as structural features such as sparsity.

REFERENCES

[1] B. Kaltenbacher and B. Hofmann. Convergence rates for the iteratively regularized GaussNewton method in Banach spaces. Inverse Problems, Vol. 26 (2010) 035007 [2] B. Kaltenbacher, A. Kirchner, and S. Veljovi´c. Goal oriented adaptivity in the IRGNM for parameter identiﬁcation in PDEs: I. reduced formulation. Inverse Problems, Vol. 30, (2014) 045001. [3] B. Kaltenbacher, A. Neubauer, O. Scherzer. Iterative Regularization Methods for Non- linear Ill-Posed Problems. Walter de Gruyter, Berlin – New York, 2008.

26 Dual weighted residual error estimation for the ﬁnite cell method

Paolo Di Stolfo Department of Mathematics, University of Salzburg, Hellbrunner Straße 34, 5020 Salzburg, Austria [email protected]

Andreas Rademacher FakultätfürMathematik (LS X), Technische UniversitätDortmund, 44221 Dortmund, Germany [email protected]

Andreas Schr¨oder Department of Mathematics, University of Salzburg, Hellbrunner Straße 34, 5020 Salzburg, Austria [email protected]

Keywords: dual weighted residual method, ﬁnite cell method, mesh adaptivity

ABSTRACT

In this talk, we present a goal-oriented error control based on the dual weighted residual method (DWR) for the ﬁnite cell method (FCM), in which the computational domain is covered by a simpler enclosing domain [1]. The error identity resulting from the DWR approach allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We use a localization technique based on a partition of unity proposed by Richter and Wick [2]. We present an adaptive strategy with the aim to balance the two error contributions. Its performance is demonstrated for linear problems in 2D with linear goal functionals.

REFERENCES

[1] J. Parvizian, A. D¨uster,E. Rank. Finite cell method. Computational Mechanics, Vo. 41 (1), 121–133, 2007. [2] T. Richter and T. Wick. Variational localizations of the dual weighted residual estimator. Journal of Computational and Applied Mathematics, Vol. 279, 192–208, 2015.

27 Adaptive Mesh Reﬁnement for Multiple Goal Functionals

B. Endtmayer Doctoral Program Computational Mathematics, Johannes Kepler University, Altenberger Straße 69, A-4040 Linz, Austria, [email protected]

T. Wick Centre de Math´ematiquesAppliqu´ees(CMAP), Ecole Polytechnique, Route de Saclay, 91128 PALAISEAU Cedex, France, [email protected]

Keywords: ﬁnite element method; mesh adaptivity; dual-weighted residual; partition-of-unity; multi-objective goal functionals; adjoint to the adjoint problem

ABSTRACT

In this presentation, we design a posteriori error estimation and mesh adaptivity for multiple goal functionals. Our method is based on a dual-weighted residual approach in which localization is achieved in a variational form using a partition-of-unity. The key advantage is that the method is simple to implement and backward integration by parts is not required. For treating multiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete error problem) and suggest an alternative way for its computation. Our algorithmic developments are substantiated for elliptic problems in terms of four different numerical tests that cover various types of challenges, such as singularities, different boundary conditions, and diverse goal functionals. Moreover, computations with higher-order finite elements are performed.

REFERENCES

[1] B. Endtmayer and T. Wick. A Partition-of-Unity Dual-Weighted Residual Approach for Multi-Objective Goal Functional Error Estimation Applied to Elliptic Problems. Com- putational Methods in Applied Mathematics, published online, doi:10.1515/cmam2017- 0001, 2017. [2] B. Endtmayer. Adaptive Mesh Reﬁnement for Multible Goal Functionals. Master thesis, Institute of Computational Mathematics, JKU Linz, 2017.

28 Residual-based a posteriori error analysis for symmetric mixed Arnold-Winther FEM

Carsten Carstensen Humboldt-Universit¨atzu Berlin, Unter den Linden 6, 10099 Berlin, Germany, [email protected]

Dietmar Gallistl Karlsruher Institut f¨urTechnologie, Englerstr. 2, 76131 Karlsruhe, Germany, [email protected]

Joscha Gedicke Universit¨atWien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria, [email protected]

Keywords: linear elasticity, mixed ﬁnite element method, a posteriori, symmetric stress ﬁnite elements

ABSTRACT

This talk introduces an explicit residual-based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after Arnold-Winther with pointwise symmetric and H(div)-conforming stress approximation. Opposed to a previous publication, the residual- based a posteriori error estimator of this talk is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence it is straightforward to implement an adaptive mesh-refining algorithm obligatory in practical computations. Numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rate for piecewise affine data is observed in the L2 stress error and repro- duced in non-smooth situations by the adaptive mesh-refining strategy.

29 On the DPG method for Signorini problems

Thomas F¨uhrer TU Wien, Wiedner Hauptstraße 8–10, 1040 Vienna, Austria, [email protected]

Norbert Heuer Pontificia Universidad Católicade Chile, VickuñaMackenna 4860, Santiago, Chile, [email protected]

Ernst P. Stephan Leibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany, [email protected]

Keywords: Contact problem, Signorini problem, variational inequality, DPG method, optimal test functions, ultra-weak formulation

ABSTRACT

We derive and analyze discontinuous Petrov-Galerkin methods with optimal test functions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and non-symmetric formulations where optimal test functions are only used for the PDE part of the problem, not the boundary conditions. For the symmetric case and lowest order approximations, we provide a simple a posteriori error estimate. In a second part, we apply our technique to the singularly perturbed case of reaction dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case

REFERENCES

[1] T. F¨uhrer,N. Heuer and E.P. Stephan. On the DPG method for Signorini problems. arXiv.org, Vol. arXiv:1609.00765, 2016.

30 Numerical methods for nonmonotone contact problems in continuum mechanics

Nina Ovcharova Universit¨atder Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, [email protected]

Keywords: non-monotone contact, regularization, non-smooth optimization, hp-adaptivity

ABSTRACT

We present several efficient numerical methods for non-convex, non-smooth variational problems in non-monotone contact. Examples include non-monotone friction and adhesive contact problems, delamination and crack propagation in adhesive bonding of composite structures. A challenging problem is adhesive bonding in case of contamination. The nonsmoothness comes from the non-smooth data of the problems itself, in particular from non-monotone, multivalued physical laws involved in the boundary conditions. The variational formulation of the resulting boundary value problems leads to a class of non-smooth variational inequalities, the so-called hemivariational inequalities (HVIs). The latter maybe viewed as a first order condition of a nonconvex, non-smooth minimization problem. These problems are much harder to analyze and solve than the classical variational inequality problems like Signorini contact or Tresca-frictional problems. The resulting HVI problem is first regularized and then discretized by either finite element or boundary element methods. In addition, we propose a novel regularized mixed formulation and provide a reliable a-posteriori error estimate enabling also hp-adaptivity. Another approach to solve nonsmooth variational problems is by the strategy: first discretize by finite (or boundary) elements, then optimize using finite dimensional non-smooth optimization methods as bundle or non-smooth trust region methods. Various numerical experiments illustrate the behavior, the strength and the limitations of the proposed approximation schemes.

REFERENCES

[1] N. Ovcharova and L. Banz. Coupling regularization and adaptive hp-BEM for the solution of a delamination problem. Numerische mathematik, DOI: 10.1007/s00211-017- 0879-5, 2017. [2] M. Dao, J. Gwinner, D. Noll, and N. Ovcharova. Nonconvex bundle method with application to a delamination problem. Computational Optimization and Applications, Vol. 65 (1), 173–203, 2016. [3] J. Gwinner and N. Ovcharova. From solvability and approximation of variational inequalities to solution of nondiﬀerentiable optimization problems in contact mechanics. Optimization, Vol. 64 (8), 1683–1702, 2015.

31 Local convergence of the boundary element method on polyhedral domains

Markus Faustmann TU Wien, Institute for Analysis and Scientiﬁc Computing, Wiedner Hauptstraße 8-10, 1040 Wien, [email protected]

Jens Markus Melenk TU Wien, Institute for Analysis and Scientiﬁc Computing, Wiedner Hauptstraße 8-10, 1040 Wien, [email protected]

Keywords: boundary element method, local error, regularity

ABSTRACT

We analyze the local behavior of the boundary element method with quasi-uniform meshes for Symm’s integral equation and the hyper-singular integral equation on polyhedral Lipschitz domains. Globally, the rate of convergence of the error is limited by the regularity of the solution, which may be reduced due to singularities of the geometry or data. However, if the quantity of interest is only a subpart of the computational domain, we can hope for better convergence behavior of the error. In this talk, we provide sharp local a-priori estimates in stronger norms (L2 and H1) than the energy norms. Thereby, the local error can be bounded by the local best-approximation error and a global error in a weak norm. Duality arguments are used to control the errors in the weak norm. They rely on elliptic shift theorems that involve both the interior and exterior problems. The numerical examples also conﬁrm the sharpness of our estimates.

32 Superconvergent graded meshes: First results

Thomas Apel Universit¨atder Bundeswehr M¨unchen, 85577 Neubiberg, Germany, [email protected]

Mariano Mateos Universidad de Oviedo, Campus de Gij´on,33203 Gij´on,Spain, [email protected]

Johannes Pfefferer TU München, Boltzmannstr. 3, 85748 Garching bei München, Germany, pfeff[email protected]

Arnd R¨osch Universt¨atDuisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany, [email protected]

Keywords: superconvergence mesh, graded mesh, error estimates

ABSTRACT

Superconvergent discretization error estimates can be obtained when the solution is smooth enough and the ﬁnite element meshes enjoy some structural properties. The simplest one is that any two adjacent triangles form a parallelogram. The solution of elliptic boundary value problems contains singularities in the vicinity of corners (and edges in 3D) leading to reduced convergence order in the case of quasi-uniform meshes. A remedy is the use of graded meshes near these corners. The question arises whether both approaches could be combined. The aim of the talk is to present ﬁrst results.

33 Polynomial robust stability analysis for H(div)-conforming ﬁnite elements for the Stokes equations

Philip Lukas Lederer Institute for Analysis an Scientiﬁc Computing - TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien , [email protected]

Joachim Sch¨oberl Institute for Analysis an Scientiﬁc Computing - TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien , [email protected]

Keywords: Navier Stokes equations, mixed ﬁnite element methods, discontinuous Galerkin methods, high order methods

ABSTRACT

In this work we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use H(div)-conforming ﬁnite elements (see [2]) as they provide major beneﬁts such as exact mass conservation and pressure-independent error estimates. The main aspect of this work lies in the analysis of high order approximations. We show that the considered method is uniformly stable with respect to the polynomial order k and provides optimal error estimates s ku − uhk1,∗ + kΠp − phk0 ≤ c(h/k) kuks+1. To derive those estimates, we prove a krobust LBB condition. This proof is based on a polynomial H2-stable extension operator. This extension operator itself is of interest for the numerical analysis of C0-continuous discontinuous Galerkin methods for 4th order problems.

REFERENCES

[1] Lederer, P. and Schöberl, J.. Joachim Polynomial robust stability analysis for H(div)- conforming finite elements for the Stokes equations. arXiv preprint arXiv:1612.01482, 2016 [2] Christoph Lehrenfeld and Joachim Schöberl. High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows Computer Methods in Applied Mechanics and Engineering, Vol. 307, 339 – 361, 2016.

34 Regularization error estimates for distributed control problems

Martin Neumüller Institut fürNumerische Mathematik, Johannes Kepler UniversitätLinz, Altenberger Str. 69, 4040 Linz, [email protected]

Olaf Steinbach Institut f¨urNumerische Mathematik, TU Graz, Steyrergasse 30, 8010 Graz, [email protected]

Keywords: Distributed control problem, regularization error

ABSTRACT

As a model problem we consider a distributed control problem with energy regularization in H−1(Ω). In the case of no control constraints the optimality system is reduced to a singularly perturbed diffusion–reaction equation. This enables us to derive regularization error estimates for the optimal state u with respect to the target u. Depending on the regularity of u we obtain different orders in the regularization parameter which is confirmed by numerical examples. We also discuss the case when the control is considered in L2(Ω).

REFERENCES

[1] M. Neum¨ullerand O. Steinbach. Regularization error estimates for distributed control problems in energy spaces. Berichte aus dem Institut f¨urNumerische Mathematik, TU Graz, 2017.

35 A Black-Box Algorithm for Fast Matrix Assembly in Isogeometric Analysis

Clemens Hofreither Department of Computational Mathematics, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria.

Keywords: Isogeometric Analysis, low-rank approximation, splines

ABSTRACT

We present a fast algorithm for assembling stiffness matrices in Isogeometric Analysis with tensor product spline spaces. The procedure exploits the facts that (a) such matrices have block-banded structure, and (b) they often have low Kronecker rank. Combined, these two properties allow us to reorder the nonzero entries of the stiffness matrix into a relatively small, dense matrix or tensor of low rank. A suitable black-box low-rank approximation algorithm is then applied to this matrix or tensor. This allows us to approximate the nonzero entries of the stiffness matrix while explicitly computing only relatively few of them, leading to a fast assembly procedure. The algorithm does not require any further knowledge of the used spline spaces, the geometry transform, or the partial differential equation, and thus is black-box in nature. Existing assembling routines can be reused with minor modifications. Numerical examples demonstrate significant speedups over a standard Gauss quadrature assem- bler for several geometries in two and three dimensions. The runtime scales sublinearly with the number of degrees of freedom in a large pre-asymptotic regime.

36 Isogeometric mortar methods in solid mechanics∗

Barbara Wohlmuth Zentrum Mathematik, Technische Universit¨atM¨unchen, Boltzmannstr. 3, 85748 Garching, [email protected]

Linus Wunderlich Zentrum Mathematik, Technische Universit¨atM¨unchen, Boltzmannstr. 3, 85748 Garching, [email protected]

∗ Funded within the DFG priority programme SPP 1748 and joint work with S. Reese, W.A. Wall and C. Wieners.

Keywords: biorthogonal basis functions, isogeometric mortar methods, vibroacoustics

ABSTRACT

The handling of multi-patch geometries is a key ingredient to practical applications of isogeometric analysis [2], as complicated domains can often not be reasonably represented by a single NURBS patch. Weak couplings of isogeometric mortar patches [1] allow for a flexible and accurate coupling of patches, without requiring the tensor product meshes to match at the interface. We highlight several solid mechanical applications, one of them is the vibroacoustical analysis of a violin bridge. The thin wooden component of a violin has an important influence on the acous- tics. The complicated and curved domain motivates the use of isogeometric mortar methods. In addition to the nine orthotropic material parameters, the thickness of the geometry is included as an extra parameter. The resulting eigenvalue problem with ten parameters is solved in a multi-query context to handle the uncertainty in the material parameter. An efficient solution in this context is guaranteed by the use of reduced basis techniques for eigenvalue problems [3]. Biorthogonal basis functions were applied in [4] for weak patch coupling as well as isogeometric discretizations of contact problems. We extend these results by constructing biorthogonal basis functions with higher order approximation properties.

REFERENCES

[1] E. Brivadis, A. Buﬀa, B. Wohlmuth and L. Wunderlich. Isogeometric mortar methods. Comput. Methods Appl. Mech. Eng., Vol. 284, 292–319, 2014. [2] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs. Isogeometric Analysis, Wiley, 2009. [3] T. Horger, B. Wohlmuth and L. Wunderlich. Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics. Accepted for publication in MOREPAS, 2015. [4] A. Seitz, P. Farah, J. Kremheller, B. Wohlmuth, W.A. Wall, A. Popp. Isogeometric dual mortar methods for computational contact mechanics. Comput. Methods Appl. Mech. Eng., Vol. 301, 259–280, 2016.

37 Functional a posteriori error estimates and adaptivity for IgA schemes

U. Langer RICAM, Austrian Academy of Sciences, Austria, Altenberger Straße 69 4040 Linz, [email protected] S. Matculevich RICAM, Austrian Academy of Sciences, Austria, Altenberger Straße 69 4040 Linz, [email protected] S. Repin St. Petersburg V.A. Steklov Institute of Mathematics, Russia, Fontanka river embankment, 27, Sankt-Peterburg, Russia, 191011, [email protected] Keywords: adaptivity for space-time IgA scheme, parabolic equation, functional type a posteriori error estimates

ABSTRACT

We are concerned with guaranteed error control of Isogeometric Analysis (IgA) numerical approximations of elliptic boundary value problems (BVPs). The approach is discussed within the d paradigm of classical linear Poisson Dirichlet model problem: ﬁnd u : Ω → R such that

−∆xu = f in Ω, u = uD on ∂Ω, (0.1)

d where Ω ⊂ R , d ∈ {1, 2, 3}, denotes a bounded domain having a Lipschitz boundary ∂Ω, ∆x is 2 1 the Laplace operator in space, f ∈ L (Ω) is a given source function, and uD ∈ H0 (Σ) is a given load on the boundary. We conduct the numerical study of the functional a posteriori error estimates integrated into the IgA framework. These so-called majorants and minorants were originally introduced in [1] and later applied to diﬀerent mathematical models. Initially, the functional approach to the error estimation in combination with IgA approximations (generated by tensor-product splines) was investigated in [2] for (0.1). In the current work, we test the algorithm of the majorant reconstruction suggested [2], which allows the considerable reduction of the time-costs for the error estimates calculation and, at the same time, generates guaranteed, sharp, and fully computable bounds of errors. Moreover, we combine functional error estimates with THB- Splines (the implementation provided by G+smo) and demonstrate their eﬃciency with respect to adaptive mesh generation in IgA schemes.

REFERENCES

[1] S. Repin. A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauch. Sem. V. A. Steklov Math. Institute in St.-Petersburg (POMI), Vol. 243, 201–214, 1997. [2] S. K. Kleiss and S. K. Tomar. Guaranteed and sharp a posteriori error estimates in isogeometric analysis. Computers & Mathematics with Applications, Vol. 70(3), 167– 190, 2015.

38 Time Discontinuous Galerkin Multipatch Isogeometric Analysis of Parabolic Diﬀusion Problems

Ioannis Toulopoulos Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences Altenbergerstr. 69, A-4040 Linz, Austria

Keywords: Parabolic initial-boundary value problems, Discontinuous Galerkin methods, Space-time Isogeometric Analysis, A priori discretization error estimates, parallel solvers.

ABSTRACT

In this talk, we present a discontinuous Galerkin (dG) time multipatch Isogeometric (IgA) scheme for the numerical solution of linear parabolic problems. We derive the weak formulation by multiplying the Partial Differential Equation (PDE) by a test function depending on spatial and time variable, and then applying integration by parts in both variables. The resulting formulation helps on deriving the analogous discrete space-time dG-IgA form. We show that, the discrete bilinear form is elliptic on the IgA space with respect to a mesh-dependent energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields a priori discretization error estimates. We present numerical results confirming the efficiency of the space-time method and the theoretical error estimates. This work was supported by the Austrian Science Fund (FWF) under the grant NFN S117-03 and is based on [1].

REFERENCES

[1] C. Hofer, U. Langer, M. Neum¨uller,and I. Toulopoulos. Multipatch time discon- tin- uous Galerkin space-time isogeometric analysis of parabolic evolution problems, under preparation, 2017.

39 Space-time boundary element methods for the heat equation

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

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

Keywords: Space-time boundary element methods, preconditioning, heat equation

ABSTRACT

In this talk we describe the boundary element method for the discretization of the timedepen- dent heat equation. In contrast to standard time-stepping schemes we consider an arbitrary decomposition of the boundary of the space-time cylinder into boundary elements, which are line segments in temporal direction in the one-dimensional case n = 1, triangles in the twodi- mensional case n = 2, and tetrahedra in the three-dimensional case n = 3. Besides adaptive refinement strategies this approach allows us to parallelize the computation of the global solution of the whole space-time system. In addition to the derivation of boundary element methods for the Dirichlet initial boundary value problem we state convergence properties and error estimates of the approximations. Those estimates are based on the approximation properties of boundary element spaces in anisotropic Sobolov spaces. The systems of linear equations are solved with the GMRES method. Based on the mapping properties of the single layer and the hypersingu- lar boundary integral operator we construct a preconditioner for the discretization of the first boundary integral equation. The theoretical results are confirmed by numerical tests.

40 Space-Time Boundary Element Method for the Wave Equation

Marco Zank Institut f¨urNumerische Mathematik, Steyrergasse 30/III, 8010 Graz, [email protected]

Keywords: Wave Equation, Boundary Element Method, Space-Time Method

ABSTRACT

For the discretisation of the wave equation by boundary element methods the starting point is the so-called Kirchhoﬀ’s formula, which is a representation formula by means of boundary potentials. In this talk diﬀerent approaches to derive weak formulations of related boundary integral equations are considered. First, a brief overview of the Laplace transform method with boundedness and coercivity estimates in appropriate Sobolev spaces is given. Second, a space- time energetic formulation is motivated and discussed. For this space-time energetic formulation a space-time boundary element method is introduced and to derive an adaptive scheme an a posteriori error estimator based on the representation formula is used. Finally, numerical examples for a one-dimensional spatial domain are presented and discussed.

41 Linear second order implicit-explicit time-integration of the (eddy-currents-)Landau-Lifshitz-Gilbert equation

Carl-Martin Pfeiler TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, [email protected]

Dirk Praetorius TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, [email protected]

Michele Ruggeri TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, [email protected]

Bernhard Stiftner TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, [email protected]

Keywords: micromagnetism, ﬁnite elements, time-marching scheme, unconditional convergence.

ABSTRACT

Combining ideas from [1] and [2], we present a numerical integrator for the integration of the Landau-Lifschitz Gilbert equation which is unconditionally convergent and formally (almost) second order in time, but requires only the solution of one linear system per time-step. Moreover, only the exchange contribution is treated implicitly in time, while the lower-order contributions like the computationally expensive stray ﬁeld are treated explicitly in time. Moreover, we extend the scheme to the coupling of the Landau-Lifschitz Gilbert equation with eddy-currents. Unlike existing integrators for this PDE system, the new integrator is unconditionally convergent and (almost) second order in time and requires only the solution of two linear systems per time-step.

REFERENCES

[1] F. Alouges, E. Kritsikis, J. Steiner and J.-C. Toussaint. A convergent and precise ﬁnite element scheme for Landau-Lifschitz-Gilbert equation. Numer. Math. 128, Vol. 3, 407–430, 2014. [2] D. Praetorius, M. Ruggeri and B. Stiftner. Convergence of an implicit-explicit midpoint scheme for computational micromagnetics. arXiv:1611.02465, 2016.

42 Time propagators for Schr¨odinger-type equations with expensive- to-evaluate nonlinear part

Harald Hofstätter Technische UniversitätWien, Institut für Theoretische Physik, Wiedner Hauptstraße 8–10, A-1040 Wien, hofi@harald-hofstaetter.at

Othmar Koch Universit¨atWien, Institut f¨urMathematik, Oskar-Morgenstern Platz 1, A-1090 Wien, [email protected]

Keywords: Time-dependent Schr¨odinger equation, Multi-Conﬁguration Method, Exponential Multistep–Lawson Method

ABSTRACT

We give an overview of numerical time integration methods for time-dependent nonlinear equations i∂tψ(t) = Aψ(t) + B(t, ψ(t)) of Schrödingertype, where the linear operator A is built up from (discretized versions of) Lapla- cians, and the nonlinear time-dependent operator B is well-behaved but expensive to evaluate. As an example we consider the equations appearing in the context of multi-configuration time- dependent Hartree–Fock (MCTDHF) calculations. We compare a number of established approaches, comprising splitting methods, composition methods, exponential Runge–Kutta methods, and exponential multistep methods. It is found that exponential predictor-corrector multistep methods of Lawson type are particularly attractive in terms of accuracy, stability, and computational effort.

List of participants

Speakers

Thomas Apel, p. 33 Clemens Hofreither, p. 36 UniBw Munich JKU Linz Germany Austria [email protected] [email protected] Boaz Blankrot, p. 17 Harald Hofstätter, p. 43 Vienna University of Technology Vienna University of Technology Austria Austria [email protected] hofi@harald-hofstaetter.at Paolo Di Stolfo, p. 27 Daniel Jodlbauer, p. 20 University of Salzburg JKU Linz Austria Austria [email protected] [email protected] Stefan Dohr, p. 40 Amirreza Khodadadian, p. 16 Graz University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Bernhard Endtmayer, p. 28 Gerhard Kitzler, p. 22 JKU Linz Vienna University of Technology Austria Austria [email protected] [email protected] Markus Faustmann, p. 32 Lukas Kogler, p. 18 Vienna University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Thomas Führer, p. 30 Philip Lukas Lederer, p. 34 Vienna University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Markus Gasteiger, p. 21 Mario Luiz Previatti de Souza, p. 26 University of Innsbruck University of Klagenfurt Austria Austria [email protected] [email protected] Joscha Gedicke, p. 29 Svetlana Matculevich, p. 38 University of Innsbruck RICAM Linz Austria Austria [email protected] [email protected] Christian Gerhards, p. 24 Gregor Mitscha-Baude, p. 13 University of Vienna Vienna University of Technology Austria Austria [email protected] [email protected]

45 Darian Onchis, p. 23 Benjamin Stadlbauer, p. 12 University of Vienna Vienna University of Technology Austria Austria [email protected] [email protected] Nina Ovcharova, p. 31 Olaf Steinbach, p. 35 UniBw Munich Graz University of Technology Germany Austria [email protected] [email protected] Gudmund Pammer, p. 15 Bernhard Stiftner, p. 42 Vienna University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Stefan Rigger, p. 14 Ioannis Toulopoulos, p. 39 Vienna University of Technology RICAM Linz Austria Austria [email protected] [email protected] Markus Sch¨obinger, p. 25 Linus Wunderlich, p. 37 Vienna University of Technology Munich University of Technology Austria Germany [email protected] [email protected] Bernd Schwarzenbacher, p. 19 Marco Zank, p. 41 Vienna University of Technology Graz University of Technology Austria Austria [email protected] [email protected]

46 Further participants

Emmanuel Akinlabi Matthias Hochsteger University of Warsaw Vienna University of Technology Poland Austria [email protected] [email protected] Niklas Angleitner Othmar Koch Vienna University of Technology University of Vienna Austria Austria [email protected] [email protected] Winfried Auzinger Raoul Kutil Vienna University of Technology University of Salzburg Austria Austria [email protected] [email protected] Lothar Banz Gregor Milicic University of Salzburg University of Salzburg Austria Austria [email protected] [email protected] Andreas Byfut Michael Neunteufel University of Salzburg Vienna University of Technology Austria Austria [email protected] [email protected] Scott Congreve G¨unther Of University of Vienna Graz University of Technology Austria Austria [email protected] [email protected] Giovanni Di Fratta Maryam Parvizi Vienna University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Kirian D¨opfner Ilaria Perugia Vienna University of Technology University of Vienna Austria Austria [email protected] [email protected] Lukas Einkemmer Jan Petsche University of Innsbruck University of Salzburg Austria Austria [email protected] [email protected] Gregor Gantner Alexander Pichler Vienna University of Technology University of Vienna Austria Austria [email protected] [email protected] Alexander Haberl Mirko Residori Vienna University of Technology University of Innsbruck Austria Austria [email protected] [email protected]

47 Alexander Rieder Joachim Sch¨oberl Vienna University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Claudio Rojik Andreas Schr¨oder Vienna University of Technology University of Salzburg Austria Austria [email protected] [email protected] Maximilian Samsinger Leila Taghizadeh University of Innsbruck Vienna University of Technology Austria Austria [email protected] [email protected] Stefan Schimanko Markus Wess Vienna University of Technology Vienna University of Technology Austria Austria [email protected] [email protected] Christoph Wintersteiger Vienna University of Technology Austria [email protected]