Hesthaven / Rønquist Spectral and High Order Methods for Partial Differential Equations

Selected papers from the ICOSAHOM '09 conference, June 22-26, Trondheim, Norway
1. Auflage 2010
ISBN: 978-3-642-15337-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 76, 510 Seiten, eBook

Reihe: Lecture Notes in Computational Science and Engineering

The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2009), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of the papers will provide the reader with a snapshot of state-of-the-art and help initiate new research directions through the extensive bibliography.

Zielgruppe

Research

Autoren/Hrsg.

Hesthaven, Jan S.

Rønquist, Einar M.

Inhaltsverzeichnis

1;Foreword;6
2;Contents;8
3;hp-FEM for the Contact Problem with Tresca Friction in Linear Elasticity: The Primal Formulation;13
3.1;1 Introduction;13
3.2;2 Problem Formulation;14
3.3;3 A Priori Error Estimates;16
3.3.1;3.1 An Interpolation Error Estimate for B1/22,1-Functions;16
3.3.2;3.2 A Polynomial Inverse Estimate;19
3.3.3;3.3 Convergence Rates: Proof of Theorem 3.1;21
3.4;4 Numerical Experiments;25
3.4.1;4.1 A Posteriori Error Estimation;25
3.4.2;4.2 Numerical Examples;26
3.5;References;28
4;On Multivariate Chebyshev Polynomials and Spectral Approximations on Triangles;30
4.1;1 Introduction;30
4.2;2 Chebyshev Polynomials and Root Systems;33
4.2.1;2.1 Root Systems;33
4.2.2;2.2 Multivariate Chebyshev Polynomials;34
4.2.3;2.3 The A2 Root System;36
4.3;3 Computing Gradients;38
4.3.1;3.1 Gradients in the A2 Root System;39
4.4;4 Clenshaw–Curtis Quadrature;41
4.4.1;4.1 Clenshaw–Curtis Quadrature in the A2 Root System;41
4.5;5 Triangles;44
4.5.1;5.1 Clenshaw–Curtis Quadrature Over a Triangle;44
4.5.2;5.2 Nonlinear Transformations;46
4.5.3;5.3 Linear Transformations;47
4.6;6 Numerics;47
4.7;7 Summary;50
4.8;References;51
5;Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison;53
5.1;1 Introduction;53
5.2;2 Problem Setting;55
5.2.1;2.1 Finite Element Approximation in the Physical Space;57
5.3;3 Polynomial Approximation in the Stochastic Dimension;57
5.3.1;3.1 Stochastic Galerkin Approximation;59
5.3.2;3.2 Stochastic Collocation Approximation on Sparse Grids;61
5.4;4 Numerical Results;64
5.4.1;4.1 Test Case 1: Isotropic Problem;64
5.4.2;4.2 Test Case 2: Anisotropic Problem;68
5.5;References;71
6;Hybridizable Discontinuous Galerkin Methods;73
6.1;1 Background;73
6.2;2 The HDG Method;75
6.2.1;2.1 The Convection-Diffusion Model Equation;75
6.2.2;2.2 Mesh and Trace Operators;76
6.2.3;2.3 Approximation Spaces;76
6.2.4;2.4 HDG Formulation;77
6.2.5;2.5 Characterization of the Numerical Trace;78
6.2.6;2.6 Relation to Other DG Methods;79
6.2.7;2.7 The Local Stabilization Parameter ;81
6.2.8;2.8 Local Postprocessing;81
6.3;3 Extensions of the Basic Algorithm;83
6.3.1;3.1 Time-Dependent Convection-Diffusion Problems;83
6.3.2;3.2 Nonlinear Convection-Diffusion Problems;84
6.3.3;3.3 Stokes Flows;85
6.3.4;3.4 Incompressible Navier–Stokes Equations;88
6.4;4 Numerical Results;90
6.5;5 Conclusions;92
6.6;References;93
7;Multivariate Modified Fourier Expansions;95
7.1;1 Introduction;95
7.2;2 The d-Variate Cube;98
7.3;3 The Hyperbolic Cross;98
7.4;4 Accelerating Convergence;99
7.4.1;4.1 The Lanczos Representation and Its Computation;99
7.4.2;4.2 The Fourier Extension Problem;100
7.5;5 The Non-Tensor Product Case;101
7.6;References;102
8;Constraint Oriented Spectral Element Method;103
8.1;1 Introduction;103
8.2;2 Constraint Oriented Polynomial Approximation;104
8.2.1;2.1 Definition and Properties;104
8.2.1.1;2.1.1 First Numerical Result;106
8.2.2;2.2 Extension to Multidimensional Case;107
8.3;3 The Constraint Oriented Effect;108
8.3.1;3.1 Numerical Results;108
8.4;References;110
9;Convergence Rates of Sparse Tensor GPC FEM for Elliptic sPDEs;111
9.1;1 Introduction;111
9.2;2 Parametrization of the Model Problem;112
9.2.1;2.1 Separation of Stochastic and Deterministic Variables;112
9.2.2;2.2 Parametric Deterministic Problem;113
9.3;3 Sparse Tensor Stochastic Galerkin Method;114
9.3.1;3.1 Sparse Tensor Galerkin Formulation;114
9.3.2;3.2 Hierarchic Discretization in L2();115
9.3.3;3.3 Hierarchic Discretization in D;116
9.3.4;3.4 Convergence Rates of Sparse Tensor sGFEM;117
9.4;4 Implementation and Numerical Examples;118
9.4.1;4.1 Localization of Quasi-Best-N-Term Coefficients;118
9.4.2;4.2 Numerical Example;118
9.5;References;119
10;A Conservative Spectral Element Method for Curvilinear Domains;121
10.1;1 Introduction;121
10.2;2 The Poisson Equation in Terms of Differential Forms;122
10.3;3 Discretization of the Transformed Poisson Equation;123
10.4;4 Results;127
10.5;5 Concluding Remarks;128
10.6;References;128
11;An Efficient Control Variate Method for Parametrized Expectations;130
11.1;1 A Control Variate Method for Parametrized Expectations;131
11.1.1;1.1 Setting of the Problem;131
11.1.2;1.2 The Control Variate Method;132
11.1.3;1.3 A Practical Approach of the Control Variate Method Deduced from Parallels with the Standard Reduced-Basis Method;133
11.2;2 Open Questions;136
11.2.1;2.1 Rigorous Certification of the Variance Reduction?;136
11.2.2;2.2 Computational Efficiency: Optimize MC Estimations?;137
11.3;References;139
12;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;140
12.1;1 Introduction;140
12.2;2 Acceleration by Conformal Mapping;142
12.2.1;2.1 Abel Extension and Conformal Mapping;142
12.2.2;2.2 Möbius Transformation and Euler Acceleration;143
12.3;3 Accelerating a Fourier Series;144
12.4;4 Numerical Illustration of Geometric Convergence;146
12.5;5 Summary;147
12.6;References;147
13;A Seamless Reduced Basis Element Method for 2D Maxwell's Problem: An Introduction;149
13.1;1 Introduction;150
13.2;2 Reduced Basis Element Method;151
13.2.1;2.1 Reduced Basis Method with Geometry As a Parameter;151
13.2.2;2.2 Reduced Basis Element Method: Formulation;155
13.2.3;2.3 Reduced Basis Element Method: Error Estimate;156
13.3;3 Numerical Results;156
13.3.1;3.1 Two-Parameter Case;157
13.3.2;3.2 Three-Parameter Case;157
13.4;4 Concluding Remarks;159
13.5;References;159
14;An hp-Nitsche's Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes;161
14.1;1 Introduction;161
14.2;2 Discretization and Notations;162
14.3;3 hp-Nitsche's Method;164
14.4;4 Quasi-Optimal Convergence;166
14.5;References;169
15;Hybrid Explicit–Implicit Time Integration for Grid-Induced Stiffness in a DGTD Method for Time Domain Electromagnetics;170
15.1;1 Introduction;170
15.2;2 Continuous Problem;171
15.3;3 Discretization in Space;172
15.4;4 Time Discretization;172
15.4.1;4.1 Explicit and Implicit Time Schemes;173
15.4.2;4.2 Hybrid Explicit–Implicit Time Scheme;174
15.5;5 Numerical Results;175
15.6;6 Conclusions;177
15.7;References;177
16;High-Order Quasi-Uniform Approximation on the Sphere Using Fourier-Finite-Elements;178
16.1;1 Introduction;178
16.2;2 Quasi-Uniform Approximation of Scalar Fields by Fourier-Finite Elements;179
16.3;3 Rotating Shallow-Water Equations;182
16.4;4 Discussion;184
16.5;References;185
17;An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations;186
17.1;1 Introduction;186
17.2;2 The hp Reduced Basis Method;188
17.3;3 A Convection-Diffusion Model Problem;191
17.4;References;193
18;Highly Accurate Discretization of the Navier–Stokes Equations in Streamfunction Formulation;195
18.1;1 Fourth Order Scheme for the Navier–Stokes Equations in Two Dimensions;195
18.2;2 The Pure Streamfunction Formulation in Three Dimensions;197
18.3;3 The Numerical Scheme;199
18.4;References;202
19;Edge Functions for Spectral Element Methods;204
19.1;1 Introduction;204
19.2;2 The Edge Functions;205
19.3;3 Application of Edge Functions to grad, curl and div;209
19.4;4 Transformations;210
19.5;5 Concluding Remarks;211
19.6;References;212
20;Modeling Effects of Electromagnetic Waves on Thin Wires with a High-Order Discontinuous Galerkin Method;213
20.1;1 Introduction;213
20.2;2 DG-FEM Discretization of Maxwell's Equations;214
20.3;3 Thin Wire Equations and DG-FEM Discretization;215
20.4;4 Field to Wire Coupling;216
20.5;5 Wire to Field Coupling;217
20.6;6 Full Field to Wire coupling;219
20.7;7 Conclusion and Outlook;221
20.8;References;221
21;A Hybrid Method for the Resolution of the Gibbs Phenomenon;223
21.1;1 Introduction;223
21.2;2 The Inverse and Statistical Filter Methods;224
21.2.1;2.1 Inverse Polynomial Reconstruction Method;224
21.2.2;2.2 Statistical Filter Method;225
21.3;3 Convergence, Accuracy and Exactness;226
21.3.1;3.1 Convergence;226
21.3.2;3.2 Covariance Matrix;227
21.3.3;3.3 Spectral Accuracy and Exactness;228
21.3.4;3.4 Numerical Convergence with Round-Off Errors;228
21.4;4 A Hybrid IPRM and SF Method: Numerical Results;229
21.5;5 Conclusions;230
21.6;References;230
22;Numerical Simulation of Fluid–Structure Interaction in Human Phonation: Verification of Structure Part;232
22.1;1 Introduction;232
22.2;2 Theory;233
22.3;3 Summation by Parts Operators;234
22.4;4 Application to Elastic Wave Equation;235
22.5;5 Discretization;236
22.6;6 Numerical Experiment;237
22.7;7 Conclusions;239
22.8;References;239
23;A New Spectral Method on Triangles;240
23.1;1 Introduction;240
23.2;2 Rectangle-to-Triangle Mapping and Nodal Basis;241
23.3;3 Implementations and Numerical Results ;245
23.4;4 Extensions and Discussions;247
23.5;References;248
24;The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case;250
24.1;1 Introduction;250
24.2;2 Offline-Online Decomposition;251
24.3;3 A Posteriori Error Estimation;254
24.4;4 Numerical Experiment;255
24.5;References;257
25;The Challenges of High Order Methods in Numerical Weather Prediction;258
25.1;1 Introduction;258
25.2;2 Overview of Atmospheric Modeling Challenges and Status;260
25.3;3 Challenges;266
25.3.1;3.1 Where High Order Holds Promise;266
25.3.2;3.2 Where High Order Instills Doubts;266
25.3.3;3.3 Recommendations;267
25.4;4 Conclusions;267
25.5;References;268
26;GMRES for Oscillatory Matrix-Valued Differential Equations;270
26.1;1 Introduction;270
26.2;2 Oscillatory Integrals;272
26.3;3 Oscillatory Differential Equations;274
26.4;4 Example: Mathieu Functions;276
26.5;References;277
27;Sensitivity Analysis of Heat Exchangers Using Perturbative Methods;278
27.1;1 Introduction;278
27.2;2 The One–Dimensional Horizontal Heat Exchanger Problem;279
27.3;3 Reference Case;281
27.4;4 Sensitivity Analysis Results;282
27.5;5 Sensitivity Analysis Accuracy;283
27.6;6 Conclusions;284
27.7;References;285
28;Spectral Element Approximation of the Hodge- Operator in Curved Elements;286
28.1;1 Introduction;286
28.2;2 Mimetic Approaches for the 2D Poisson Equation;287
28.3;3 Weak Material Laws: The Role of Least-Squares;288
28.4;4 Application to the 2D Poisson Equation;289
28.4.1;4.1 Straight Elements;289
28.4.2;4.2 Curved Elements;289
28.4.2.1;4.2.1 The Inner Product;290
28.4.2.2;4.2.2 The Hodge- Operator;291
28.4.2.3;4.2.3 The Least-Squares Residual;292
28.5;5 Concluding Remarks;292
28.6;References;293
29;Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods;295
29.1;1 Mathematical Framework;296
29.1.1;1.1 SLC in a Nutshell;296
29.1.2;1.2 gPC in a Nutshell;296
29.2;2 Application of sG-gPC to the p-System in Lagrangian Coordinates;297
29.2.1;2.1 Closure of (6) or Treatment of Non Linearities;299
29.2.2;2.2 Discontinuous Solutions and Gibbs Phenomenon;301
29.3;3 The Intrusive Polynomial Moment Method (IPMM);301
29.3.1;3.1 Analogy with Kt and Mt for the Closure;302
29.4;4 Numerical Tests;304
29.4.1;4.1 Comparison Between sG-gPC and IPMM: Burgers;304
29.4.2;4.2 Stochastic Riemann Problem: Euler System;305
29.5;5 Conclusions;306
29.6;References;307
30;Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map;308
30.1;1 Introduction;308
30.2;2 Reduced Basis Approximation of Parametric Advection-Diffusion Equations;309
30.3;3 Numerical Example;313
30.4;4 Conclusions;314
30.5;References;315
31;Constrained Approximation in hp-FEM: Unsymmetric Subdivisions and Multi-Level Hanging Nodes;317
31.1;1 Introduction;317
31.2;2 Tensor Product Shape Functions of Legendre Type;318
31.3;3 Constraints Coefficients and Multi-Level Hanging Nodes;321
31.4;4 Numerical Results;323
31.5;References;324
32;High Order Filter Methods for Wide Range of Compressible Flow Speeds;326
32.1;1 Original High Order Filter Method;326
32.2;2 Improved High Order Filter Method;328
32.3;3 Numerical Results;330
32.3.1;3.1 1-D Shock/Turbulence Interaction Problem;331
32.3.2;3.2 Taylor–Green Vortex;332
32.3.3;3.3 Compressible Isotropic Turbulence with Shocklets;333
32.4;4 New Flow Sensor for a Wide Spectrum of Flow Speedand Shock Strength;334
32.5;References;335
33;hp-Adaptive CEM in Practical Applications;337
33.1;1 Introduction;337
33.2;2 The Scattering Matrix Based Approach ;338
33.3;3 Results;340
33.3.1;3.1 Fully Coupled Interior and Exterior Model;340
33.3.2;3.2 Scattering Matrix Based Interior Only Model;342
33.4;References;344
34;Anchor Points Matter in ANOVA Decomposition;345
34.1;1 Introduction;345
34.2;2 Weights and Effective Dimension;346
34.3;3 Numerical Examples;351
34.4;References;353
35;An Explicit Discontinuous Galerkin Scheme with Divergence Cleaning for Magnetohydrodynamics;354
35.1;1 Introduction;354
35.2;2 STE-DG Discretization;355
35.2.1;2.1 Space-Time Expansion;355
35.2.2;2.2 Local Time Stepping;356
35.3;3 Divergence Correction and Local Time Stepping;356
35.4;4 Numerical Results;359
35.4.1;4.1 Convergence Test;359
35.4.2;4.2 Orszag–Tang Vortex;359
35.5;5 Conclusions;360
35.6;References;361
36;High Order Polynomial Interpolation of Parameterized Curves;362
36.1;1 Introduction;362
36.2;2 Interpolation Methods for Plane Curves;363
36.2.1;2.1 Common Interpolation Methods;364
36.2.2;2.2 The L2-Method;364
36.2.3;2.3 The Equal-Tangent Method;365
36.2.4;2.4 Numerical Results;365
36.3;3 Interpolation of Space Curves;366
36.3.1;3.1 The L2-Method;366
36.3.2;3.2 The Equal-Tangent Method;367
36.3.3;3.3 Numerical Results;367
36.4;4 Conclusions and Future Work;368
36.5;References;369
37;A New Discontinuous Galerkin Method for the Navier–Stokes Equations;370
37.1;1 Introduction;370
37.2;2 Numerical Discretization;371
37.2.1;2.1 DG Formulation and Time Stepping;371
37.2.2;2.2 The Elastoplast Method (EDG);372
37.2.2.1;2.2.1 DG Basis and Implementation;373
37.3;3 Numerical Results;373
37.3.1;3.1 1D Diffusion;374
37.3.2;3.2 Couette Thermal Flow;374
37.3.3;3.3 Blasius Boundary Layer;375
37.3.4;3.4 Supersonic Mixing Layer;376
37.4;4 Conclusions;377
37.5;References;377
38;A Pn,-Based Method for Linear Nonconstant Coefficients High Order Eigenvalue Problems;379
38.1;1 Introduction;379
38.2;2 Physical and Mathematical Preliminaries;381
38.3;3 Numerical Results;383
38.3.1;3.1 The Rigid Boundaries Case;383
38.3.2;3.2 The Free Boundary Case;384
38.4;4 Conclusions;386
38.5;References;387
39;Spectral Element Discretization of Optimal Control Problems;388
39.1;1 Linear Optimal Control Problem;389
39.2;2 SEM Discretization;390
39.3;3 Iteration and Discretization Error Estimates;392
39.4;4 Numerical Results;394
39.5;5 Conclusions;394
39.6;References;396
40;Applications of High Order Methods to Vortex Instability Calculations;397
40.1;1 Introduction;397
40.2;2 Theory;399
40.2.1;2.1 The Basic Flows;399
40.2.2;2.2 The BiGlobal Eigenvalue Problem (EVP);400
40.3;3 Results;401
40.3.1;3.1 Basic Flow;401
40.3.2;3.2 Instability Analyses;401
40.4;4 Conclusions and Outlook;403
40.5;References;404
41;Entropy Viscosity Method for High-Order Approximations of Conservation Laws;405
41.1;1 Introduction;405
41.2;2 The Entropy Viscosity Method;406
41.3;3 2D Burgers (Fourier);408
41.4;4 KPP Rotating Wave (SEM);409
41.5;5 2D Euler System (Fourier);411
41.6;References;412
42;High-Order Accurate Numerical Solution of Incompressible Slip Flow and Heat Transfer in Microchannels;413
42.1;1 Introduction;413
42.2;2 Problem Formulation;414
42.3;3 Wall Boundary Conditions;415
42.4;4 Numerical Procedure;415
42.5;5 Numerical Results and Discussion;416
42.6;6 Concluding Remarks;420
42.7;References;421
43;Spectral Methods for Time-Dependent Variable-Coefficient PDE Based on Block Gaussian Quadrature;422
43.1;1 Introduction;422
43.2;2 Krylov Subspace Spectral Methods;423
43.3;3 Implementation;426
43.4;4 Application to Maxwell's Equations;426
43.5;5 Numerical Results;428
43.5.1;5.1 Parabolic Problems;428
43.5.2;5.2 Maxwell's Equations;429
43.6;6 Summary and Future Work;431
43.7;References;431
44;The Spectral Element Method Used to Assess the Quality of a Global C1 Map;433
44.1;1 Introduction;433
44.2;2 Methods;434
44.3;3 Regularity;439
44.4;References;440
45;Stabilization of the Spectral-Element Method in Turbulent Flow Simulations;441
45.1;1 Introduction;441
45.2;2 Equations and Discretization;442
45.3;3 Stabilization of Turbulent Flow Simulations;443
45.4;4 Analysis of Model Problems;443
45.4.1;4.1 1D: Stabilization of the Burgers' Equation;443
45.4.2;4.2 2D: Recovery of Skew-Symmetry for the SEM Convection Operator in the Scalar Transport Equation;445
45.5;5 Application to the Navier–Stokes Equations;446
45.5.1;5.1 3D: Subcritical K-type Transition Simulations;447
45.5.2;5.2 3D: Fully Turbulent Channel Flow Simulationsat Re = 590;448
45.6;6 Conclusions;449
45.7;References;449
46;The Spectral-Element and Pseudo-Spectral Methods: A Comparative Study;451
46.1;1 Introduction;451
46.2;2 Study Setup;452
46.3;3 Results;453
46.3.1;3.1 Part A: Efficiency;453
46.3.2;3.2 Part B: Accuracy in Transitional Flow Simulations;454
46.3.3;3.3 Part B: Accuracy in Turbulent Flow Simulations;455
46.4;4 Conclusions;457
46.5;References;458
47;Adaptive Spectral Filtering and Digital Total Variation Postprocessing for the DG Method on Triangular Grids: Application to the Euler Equations;460
47.1;1 Introduction;460
47.2;2 The Discontinuous Galerkin Scheme with Spectral Filtering;461
47.3;3 The Digital Total Variation Filter;462
47.4;4 Numerical Experiments;463
47.5;References;467
48;BDDC and FETI-DP Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity;469
48.1;1 Introduction;469
48.2;2 Almost Incompressible Elasticity and Spectral Elements;469
48.3;3 The BDDC Algorithm;471
48.4;4 Numerical Results in the Plane;474
48.5;References;475
49;A Two-Dimensional DG-SEM Approach to Investigate Resonance Frequencies and Sound Radiation of Woodwind Instruments;477
49.1;1 Introduction;477
49.2;2 Discontinuous Galerkin Method for the Euler Equations;479
49.2.1;2.1 Conservation Equations;479
49.2.2;2.2 Numerical Scheme;479
49.3;3 The Influence of the Vocal Tract on the Recorder;480
49.3.1;3.1 Problem Description;480
49.3.2;3.2 Influence of the Vocal Tract;481
49.4;4 Sound Radiation of the Bassoon;482
49.5;5 Conclusions;484
49.6;References;484
50;Spectral Properties of Discontinuous Galerkin Space Operators on Curved Meshes;485
50.1;1 Introduction;485
50.2;2 Method;486
50.2.1;2.1 Discontinuous Galerkin Method;486
50.2.2;2.2 Stability Analysis;487
50.3;3 Results;488
50.3.1;3.1 Qualitative Results in 2D;489
50.3.2;3.2 Dependence on the Local Jacobian in 1D;490
50.3.3;3.3 Estimation Based on Integration Matrices in 1D;491
50.4;4 Conclusions;491
50.5;References;492
51;Post-Processing of Marginally Resolved Spectral Element Data;493
51.1;1 Introduction;493
51.2;2 Numerical Test Problems;494
51.2.1;2.1 An Analytical Example;494
51.2.2;2.2 Turbulent Channel Flow;494
51.3;3 Interface Averaging;495
51.4;4 Improved Interface Treatment;497
51.4.1;4.1 Polynomial Interpolation;497
51.4.2;4.2 Filtering;499
51.5;5 Conclusions;499
51.6;References;500
52;Editorial Policy;501
53;Lecture Notes in Computational Science and Engineering;503
54;Monographs in Computational Science and Engineering;507
55;Texts in Computational Science and Engineering;507