Lecture Notes in Computational Science and Engineering 76

Editors: Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick

For further volumes: http://www.springer.com/series/3527 • Jan S. Hesthaven • Einar M. Rønquist Editors

Spectral and High Order Methods for Partial Differential Equations

Selected papers from the ICOSAHOM ’09 conference, June 22-26, Trondheim, Norway

123 Editors Jan S. Hesthaven Einar M. Rønquist Brown University Norwegian University of Science Division of Applied Mathematics and Technology 182 George Street Department of Mathematical Sciences Providence, RI 02912 7491 Trondheim USA Norway [email protected] [email protected]

ISBN 978-3-642-15336-5 e-ISBN 978-3-642-15337-2 DOI 10.1007/978-3-642-15337-2 Springer Heidelberg Dordrecht London New York

Mathematics Subject Classification (2010): 65M70, 65N35, 65N30, 74S25, 76M10, 76M22, 78M10, 78M22

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Cover illustration: The nudg++ team - Tim Warburton (Rice University), Nigel Nunn, Nico Gödel (Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg)

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Springer is part of Springer Science+Business Media (www.springer.com) Foreword

This volume presents selected papers from the eigth ICOSAHOM (International Conference On Spectral and High Order Methods) conference which was held at the Norwegian University of Science and Technology (NTNU), Trondheim, Norway, during the week June 22–26, 2009. These papers were refereed by members of the scientific committee of ICOSAHOM as well as by other leading scientists. The first ICOSAHOM conference was held in Como, Italy, in 1989. At that point the importance of high order methods was recognized and it was deciced to organize a series of conferences to be held every 3 years (Montpelier, France, 1992; Houston, TX, USA, 1995; Tel Aviv, Israel, 1998; Uppsala, Sweden, 2001; Providence, RI, USA, 2004; and Beijing, China, 2007). At the conference in Beijing in 2007, it was decided to organize the conferences every other year. This decision was made partly due to the growing importance and the growing activity in this field. From the interest seen at ICOSAHOM 2009, this seems to have been an appropriate decision. The number of registered participants was over 200, while the total number of talks was 215, comprising nine invited talks, 153 talks in 19 different topic-specific minisymposia, and 53 talks in various contributed sessions. The ICOSAHOM conferences remain the main meeting place for researchers with interest in the theoretical, applied and computational aspects of high order methods for the numerical solution of partial differential equations. The content of the proceedings is organized as follows. First, contributions from the invited speakers are included, listed in alphabetical order according to the invited speaker. Next, contributions from the speakers at all the minisymposia are included, listed in alphabetical order according to the first author of each paper. Finally, con- tributions from the speakers at the various contributed sessions are included, also listed in alphabetical order according to the the first author. As part of the conference, a special minisymposium was organized in memory of David Gottlieb who passed away in December 2008, and who left an indelible mark on the field of applied mathematics in general and spectral methods in particular. The success of the meeting was ensured through the generous financial support by the Research Council of Norway, the National Science Foundation (NSF), the Norwegian University of Science and Technology (NTNU – through the Faculty of Information Technology, Mathematics and Electrical Engineering and through

v vi Foreword the Program for Computational Science and Visualization), and Simula Research Laboratory (through Center for Biomedical Computing). Finally, the conference could not have happened without the invaluable support and assistance of conference coordinator Anne Kajander. Special thanks also go to Tormod Bjøntegaard for all his contributions. The assistance from the members of the Numerical Analysis Group and from the graduate students at the Department of Mathematical Sciences at NTNU is also gratefully acknowledged.

Jan S. Hesthaven Einar M. Rønquist Contents

hp-FEM for the Contact Problem with Tresca Friction in Linear Elasticity: The Primal Formulation ...... 1 P. Dorsek ¨ and J.M. Melenk

On Multivariate Chebyshev Polynomials and Spectral Approximations on Triangles ...... 19 Brett N. Ryland and Hans Z. Munthe-Kaas

Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison ...... 43 Joakim B¨ack, Fabio Nobile, Lorenzo Tamellini, and Raul Tempone

Hybridizable Discontinuous Galerkin Methods ...... 63 N.C. Nguyen, J. Peraire, and B. Cockburn

Multivariate Modified Fourier Expansions ...... 85 Ben Adcock and Daan Huybrechs

Constraint Oriented Spectral Element Method ...... 93 E. Ahusborde, M. Aza¨ıez, and R. Gruber

Convergence Rates of Sparse Tensor GPC FEM for Elliptic sPDEs ...... 101 Marcel Bieri, Roman Andreev, and Christoph Schwab

A Conservative Spectral Element Method for Curvilinear Domains ...... 111 Mick Bouman, Artur Palha, Jasper Kreeft, and Marc Gerritsma

An Efficient Control Variate Method for Parametrized Expectations ...... 121 S´ebastien Boyaval

vii viii Contents

A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon ...... 131 John P. Boyd

A Seamless Reduced Basis Element Method for 2D Maxwell’s Problem: An Introduction...... 141 Yanlai Chen, Jan S. Hesthaven, and Yvon Maday

An hp-Nitsche’s Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes...... 153 Alexey Chernov and Peter Hansbo

Hybrid ExplicitÐImplicit Time Integration for Grid- Induced Stiffness in a DGTD Method for Time Domain Electromagnetics ...... 163 Victorita Dolean, Hassan Fahs, Loula Fezoui, and St´ephane Lanteri

High-Order Quasi-Uniform Approximation on the Sphere Using Fourier-Finite-Elements ...... 171 T. Dubos

An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations ...... 179 Jens L. Eftang, Anthony T. Patera, and Einar M. Rønquist

Highly Accurate Discretization of the NavierÐStokes Equations in Streamfunction Formulation...... 189 D. Fishelov, M. Ben-Artzi, and J.-P. Croisille

Edge Functions for Spectral Element Methods ...... 199 Marc Gerritsma

Modeling Effects of Electromagnetic Waves on Thin Wires with a High-Order Discontinuous Galerkin Method...... 209 N. G¨odel, T. Warburton, and M. Clemens

A Hybrid Method for the Resolution of the Gibbs Phenomenon ...... 219 Jae-Hun Jung

Numerical Simulation of FluidÐStructure Interaction in Human Phonation: Verification of Structure Part ...... 229 Martin Larsson and Bernhard M¨uller Contents ix

A New Spectral Method on Triangles ...... 237 Youyun Li, Li-Lian Wang, Huiyuan Li, and Heping Ma

The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case ...... 247 A.E. Løvgren, Y. Maday, and E.M. Rønquist

The Challenges of High Order Methods in Numerical Weather Prediction ...... 255 Catherine Mavriplis

GMRES for Oscillatory Matrix-Valued Differential Equations ...... 267 Sheehan Olver

Sensitivity Analysis of Heat Exchangers Using Perturbative Methods ...... 275 J.C. Pacio, C.A. Dorao, and M. Fernandino

Spectral Element Approximation of the Hodge-? Operator in Curved Elements ...... 283 Artur Palha and Marc Gerritsma

Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods ...... 293 G. Po¨ette, B. Despr´es, and D. Lucor

Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map ...... 307 Gianluigi Rozza, Toni Lassila, and Andrea Manzoni

Constrained Approximation in hp-FEM: Unsymmetric Subdivisions and Multi-Level Hanging Nodes ...... 317 Andreas Schr¨oder

High Order Filter Methods for Wide Range of Compressible Flow Speeds ...... 327 H.C. Yee and Bj¨orn Sj¨ogreen hp-Adaptive CEM in Practical Applications ...... 339 Adam Zdunek and Waldemar Rachowicz

Anchor Points Matter in ANOVA Decomposition ...... 347 Zhongqiang Zhang, Minseok Choi, and George Em Karniadakis x Contents

An Explicit Discontinuous Galerkin Scheme with Divergence Cleaning for Magnetohydrodynamics ...... 357 Christoph Altmann

High Order Polynomial Interpolation of Parameterized Curves ...... 365 Tormod Bjøntegaard, Einar M. Rønquist, and Øystein Tr˚asdahl

A New Discontinuous Galerkin Method for the NavierÐStokes Equations ...... 373 M. Borrel and J. Ryan

˛;ˇ A Pn -Based Method for Linear Nonconstant Coefficients High Order Eigenvalue Problems ...... 383 F.I. Dragomirescu

Spectral Element Discretization of Optimal Control Problems ...... 393 Loredana Gaudio and Alfio Quarteroni

Applications of High Order Methods to Vortex Instability Calculations ...... 403 Leo M. Gonz´alez, Vassilis Theofilis, and Fernando Meseguer-Garrido

Entropy Viscosity Method for High-Order Approximations of Conservation Laws...... 411 J.L. Guermond and R. Pasquetti

High-Order Accurate Numerical Solution of Incompressible Slip Flow and Heat Transfer in Microchannels...... 419 Kazem Hejranfar, Mir Hamed Mohafez, and Ali Khajeh-Saeed

Spectral Methods for Time-Dependent Variable-Coefficient PDE Based on Block Gaussian Quadrature...... 429 James V. Lambers

The Spectral Element Method Used to Assess the Quality of a Global C 1 Map ...... 441 A.E. Løvgren, Y. Maday, and E.M. Rønquist

Stabilization of the Spectral-Element Method in Turbulent Flow Simulations ...... 449 J. Ohlsson, P. Schlatter, P.F. Fischer, and D.S. Henningson Contents xi

The Spectral-Element and Pseudo-Spectral Methods: A Comparative Study ...... 459 J. Ohlsson, P. Schlatter, C. Mavriplis, and D.S. Henningson

Adaptive Spectral Filtering and Digital Total Variation Postprocessing for the DG Method on Triangular Grids: Application to the Euler Equations ...... 469 S. Ortleb, A. Meister, and Th. Sonar

BDDC and FETI-DP Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity ...... 479 Luca F. Pavarino and Olof B. Widlund

A Two-Dimensional DG-SEM Approach to Investigate Resonance Frequencies and Sound Radiation of Woodwind Instruments ...... 487 Andreas Richter and J¨org Stiller

Spectral Properties of Discontinuous Galerkin Space Operators on Curved Meshes...... 495 Thomas Toulorge and Wim Desmet

Post-Processing of Marginally Resolved Spectral Element Data ...... 503 Carl Erik Wasberg