Spectral and High Order Methods for Partial Differential Equations (eBook)

Foreword 6
Contents 8
hp-FEM for the Contact Problem with Tresca Friction in Linear Elasticity: The Primal Formulation 13
1 Introduction 13
2 Problem Formulation 14
3 A Priori Error Estimates 16
3.1 An Interpolation Error Estimate for B1/22,1-Functions 16
3.2 A Polynomial Inverse Estimate 19
3.3 Convergence Rates: Proof of Theorem 3.1 21
4 Numerical Experiments 25
4.1 A Posteriori Error Estimation 25
4.2 Numerical Examples 26
References 28
On Multivariate Chebyshev Polynomials and Spectral Approximations on Triangles 30
1 Introduction 30
2 Chebyshev Polynomials and Root Systems 33
2.1 Root Systems 33
2.2 Multivariate Chebyshev Polynomials 34
2.3 The A2 Root System 36
3 Computing Gradients 38
3.1 Gradients in the A2 Root System 39
4 Clenshaw–Curtis Quadrature 41
4.1 Clenshaw–Curtis Quadrature in the A2 Root System 41
5 Triangles 44
5.1 Clenshaw–Curtis Quadrature Over a Triangle 44
5.2 Nonlinear Transformations 46
5.3 Linear Transformations 47
6 Numerics 47
7 Summary 50
References 51
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison 53
1 Introduction 53
2 Problem Setting 55
2.1 Finite Element Approximation in the Physical Space 57
3 Polynomial Approximation in the Stochastic Dimension 57
3.1 Stochastic Galerkin Approximation 59
3.2 Stochastic Collocation Approximation on Sparse Grids 61
4 Numerical Results 64
4.1 Test Case 1: Isotropic Problem 64
4.2 Test Case 2: Anisotropic Problem 68
References 71
Hybridizable Discontinuous Galerkin Methods 73
1 Background 73
2 The HDG Method 75
2.1 The Convection-Diffusion Model Equation 75
2.2 Mesh and Trace Operators 76
2.3 Approximation Spaces 76
2.4 HDG Formulation 77
2.5 Characterization of the Numerical Trace 78
2.6 Relation to Other DG Methods 79
2.7 The Local Stabilization Parameter 81
2.8 Local Postprocessing 81
3 Extensions of the Basic Algorithm 83
3.1 Time-Dependent Convection-Diffusion Problems 83
3.2 Nonlinear Convection-Diffusion Problems 84
3.3 Stokes Flows 85
3.4 Incompressible Navier–Stokes Equations 88
4 Numerical Results 90
5 Conclusions 92
References 93
Multivariate Modified Fourier Expansions 95
1 Introduction 95
2 The d-Variate Cube 98
3 The Hyperbolic Cross 98
4 Accelerating Convergence 99
4.1 The Lanczos Representation and Its Computation 99
4.2 The Fourier Extension Problem 100
5 The Non-Tensor Product Case 101
References 102
Constraint Oriented Spectral Element Method 103
1 Introduction 103
2 Constraint Oriented Polynomial Approximation 104
2.1 Definition and Properties 104
2.1.1 First Numerical Result 106
2.2 Extension to Multidimensional Case 107
3 The Constraint Oriented Effect 108
3.1 Numerical Results 108
References 110
Convergence Rates of Sparse Tensor GPC FEM for Elliptic sPDEs 111
1 Introduction 111
2 Parametrization of the Model Problem 112
2.1 Separation of Stochastic and Deterministic Variables 112
2.2 Parametric Deterministic Problem 113
3 Sparse Tensor Stochastic Galerkin Method 114
3.1 Sparse Tensor Galerkin Formulation 114
3.2 Hierarchic Discretization in L2() 115
3.3 Hierarchic Discretization in D 116
3.4 Convergence Rates of Sparse Tensor sGFEM 117
4 Implementation and Numerical Examples 118
4.1 Localization of Quasi-Best-N-Term Coefficients 118
4.2 Numerical Example 118
References 119
A Conservative Spectral Element Method for Curvilinear Domains 121
1 Introduction 121
2 The Poisson Equation in Terms of Differential Forms 122
3 Discretization of the Transformed Poisson Equation 123
4 Results 127
5 Concluding Remarks 128
References 128
An Efficient Control Variate Method for Parametrized Expectations 130
1 A Control Variate Method for Parametrized Expectations 131
1.1 Setting of the Problem 131
1.2 The Control Variate Method 132
1.3 A Practical Approach of the Control Variate Method Deduced from Parallels with the Standard Reduced-Basis Method 133
2 Open Questions 136
2.1 Rigorous Certification of the Variance Reduction? 136
2.2 Computational Efficiency: Optimize MC Estimations? 137
References 139
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
1 Introduction 140
2 Acceleration by Conformal Mapping 142
2.1 Abel Extension and Conformal Mapping 142
2.2 Möbius Transformation and Euler Acceleration 143
3 Accelerating a Fourier Series 144
4 Numerical Illustration of Geometric Convergence 146
5 Summary 147
References 147
A Seamless Reduced Basis Element Method for 2D Maxwell's Problem: An Introduction 149
1 Introduction 150
2 Reduced Basis Element Method 151
2.1 Reduced Basis Method with Geometry As a Parameter 151
2.2 Reduced Basis Element Method: Formulation 155
2.3 Reduced Basis Element Method: Error Estimate 156
3 Numerical Results 156
3.1 Two-Parameter Case 157
3.2 Three-Parameter Case 157
4 Concluding Remarks 159
References 159
An hp-Nitsche's Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes 161
1 Introduction 161
2 Discretization and Notations 162
3 hp-Nitsche's Method 164
4 Quasi-Optimal Convergence 166
References 169
Hybrid Explicit–Implicit Time Integration for Grid-Induced Stiffness in a DGTD Method for Time Domain Electromagnetics 170
1 Introduction 170
2 Continuous Problem 171
3 Discretization in Space 172
4 Time Discretization 172
4.1 Explicit and Implicit Time Schemes 173
4.2 Hybrid Explicit–Implicit Time Scheme 174
5 Numerical Results 175
6 Conclusions 177
References 177
High-Order Quasi-Uniform Approximation on the Sphere Using Fourier-Finite-Elements 178
1 Introduction 178
2 Quasi-Uniform Approximation of Scalar Fields by Fourier-Finite Elements 179
3 Rotating Shallow-Water Equations 182
4 Discussion 184
References 185
An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations 186
1 Introduction 186
2 The hp Reduced Basis Method 188
3 A Convection-Diffusion Model Problem 191
References 193
Highly Accurate Discretization of the Navier–Stokes Equations in Streamfunction Formulation 195
1 Fourth Order Scheme for the Navier–Stokes Equations in Two Dimensions 195
2 The Pure Streamfunction Formulation in Three Dimensions 197
3 The Numerical Scheme 199
References 202
Edge Functions for Spectral Element Methods 204
1 Introduction 204
2 The Edge Functions 205
3 Application of Edge Functions to grad, curl and div 209
4 Transformations 210
5 Concluding Remarks 211
References 212
Modeling Effects of Electromagnetic Waves on Thin Wires with a High-Order Discontinuous Galerkin Method 213
1 Introduction 213
2 DG-FEM Discretization of Maxwell's Equations 214
3 Thin Wire Equations and DG-FEM Discretization 215
4 Field to Wire Coupling 216
5 Wire to Field Coupling 217
6 Full Field to Wire coupling 219
7 Conclusion and Outlook 221
References 221
A Hybrid Method for the Resolution of the Gibbs Phenomenon 223
1 Introduction 223
2 The Inverse and Statistical Filter Methods 224
2.1 Inverse Polynomial Reconstruction Method 224
2.2 Statistical Filter Method 225
3 Convergence, Accuracy and Exactness 226
3.1 Convergence 226
3.2 Covariance Matrix 227
3.3 Spectral Accuracy and Exactness 228
3.4 Numerical Convergence with Round-Off Errors 228
4 A Hybrid IPRM and SF Method: Numerical Results 229
5 Conclusions 230
References 230
Numerical Simulation of Fluid–Structure Interaction in Human Phonation: Verification of Structure Part 232
1 Introduction 232
2 Theory 233
3 Summation by Parts Operators 234
4 Application to Elastic Wave Equation 235
5 Discretization 236
6 Numerical Experiment 237
7 Conclusions 239
References 239
A New Spectral Method on Triangles 240
1 Introduction 240
2 Rectangle-to-Triangle Mapping and Nodal Basis 241
3 Implementations and Numerical Results 245
4 Extensions and Discussions 247
References 248
The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case 250
1 Introduction 250
2 Offline-Online Decomposition 251
3 A Posteriori Error Estimation 254
4 Numerical Experiment 255
References 257
The Challenges of High Order Methods in Numerical Weather Prediction 258
1 Introduction 258
2 Overview of Atmospheric Modeling Challenges and Status 260
3 Challenges 266
3.1 Where High Order Holds Promise 266
3.2 Where High Order Instills Doubts 266
3.3 Recommendations 267
4 Conclusions 267
References 268
GMRES for Oscillatory Matrix-Valued Differential Equations 270
1 Introduction 270
2 Oscillatory Integrals 272
3 Oscillatory Differential Equations 274
4 Example: Mathieu Functions 276
References 277
Sensitivity Analysis of Heat Exchangers Using Perturbative Methods 278
1 Introduction 278
2 The One–Dimensional Horizontal Heat Exchanger Problem 279
3 Reference Case 281
4 Sensitivity Analysis Results 282
5 Sensitivity Analysis Accuracy 283
6 Conclusions 284
References 285
Spectral Element Approximation of the Hodge- Operator in Curved Elements 286
1 Introduction 286
2 Mimetic Approaches for the 2D Poisson Equation 287
3 Weak Material Laws: The Role of Least-Squares 288
4 Application to the 2D Poisson Equation 289
4.1 Straight Elements 289
4.2 Curved Elements 289
4.2.1 The Inner Product 290
4.2.2 The Hodge- Operator 291
4.2.3 The Least-Squares Residual 292
5 Concluding Remarks 292
References 293
Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods 295
1 Mathematical Framework 296
1.1 SLC in a Nutshell 296
1.2 gPC in a Nutshell 296
2 Application of sG-gPC to the p-System in Lagrangian Coordinates 297
2.1 Closure of (6) or Treatment of Non Linearities 299
2.2 Discontinuous Solutions and Gibbs Phenomenon 301
3 The Intrusive Polynomial Moment Method (IPMM) 301
3.1 Analogy with Kt and Mt for the Closure 302
4 Numerical Tests 304
4.1 Comparison Between sG-gPC and IPMM: Burgers 304
4.2 Stochastic Riemann Problem: Euler System 305
5 Conclusions 306
References 307
Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map 308
1 Introduction 308
2 Reduced Basis Approximation of Parametric Advection-Diffusion Equations 309
3 Numerical Example 313
4 Conclusions 314
References 315
Constrained Approximation in hp-FEM: Unsymmetric Subdivisions and Multi-Level Hanging Nodes 317
1 Introduction 317
2 Tensor Product Shape Functions of Legendre Type 318
3 Constraints Coefficients and Multi-Level Hanging Nodes 321
4 Numerical Results 323
References 324
High Order Filter Methods for Wide Range of Compressible Flow Speeds 326
1 Original High Order Filter Method 326
2 Improved High Order Filter Method 328
3 Numerical Results 330
3.1 1-D Shock/Turbulence Interaction Problem 331
3.2 Taylor–Green Vortex 332
3.3 Compressible Isotropic Turbulence with Shocklets 333
4 New Flow Sensor for a Wide Spectrum of Flow Speedand Shock Strength 334
References 335
hp-Adaptive CEM in Practical Applications 337
1 Introduction 337
2 The Scattering Matrix Based Approach 338
3 Results 340
3.1 Fully Coupled Interior and Exterior Model 340
3.2 Scattering Matrix Based Interior Only Model 342
References 344
Anchor Points Matter in ANOVA Decomposition 345
1 Introduction 345
2 Weights and Effective Dimension 346
3 Numerical Examples 351
References 353
An Explicit Discontinuous Galerkin Scheme with Divergence Cleaning for Magnetohydrodynamics 354
1 Introduction 354
2 STE-DG Discretization 355
2.1 Space-Time Expansion 355
2.2 Local Time Stepping 356
3 Divergence Correction and Local Time Stepping 356
4 Numerical Results 359
4.1 Convergence Test 359
4.2 Orszag–Tang Vortex 359
5 Conclusions 360
References 361
High Order Polynomial Interpolation of Parameterized Curves 362
1 Introduction 362
2 Interpolation Methods for Plane Curves 363
2.1 Common Interpolation Methods 364
2.2 The L2-Method 364
2.3 The Equal-Tangent Method 365
2.4 Numerical Results 365
3 Interpolation of Space Curves 366
3.1 The L2-Method 366
3.2 The Equal-Tangent Method 367
3.3 Numerical Results 367
4 Conclusions and Future Work 368
References 369
A New Discontinuous Galerkin Method for the Navier–Stokes Equations 370
1 Introduction 370
2 Numerical Discretization 371
2.1 DG Formulation and Time Stepping 371
2.2 The Elastoplast Method (EDG) 372
2.2.1 DG Basis and Implementation 373
3 Numerical Results 373
3.1 1D Diffusion 374
3.2 Couette Thermal Flow 374
3.3 Blasius Boundary Layer 375
3.4 Supersonic Mixing Layer 376
4 Conclusions 377
References 377
A Pn,-Based Method for Linear Nonconstant Coefficients High Order Eigenvalue Problems 379
1 Introduction 379
2 Physical and Mathematical Preliminaries 381
3 Numerical Results 383
3.1 The Rigid Boundaries Case 383
3.2 The Free Boundary Case 384
4 Conclusions 386
References 387
Spectral Element Discretization of Optimal Control Problems 388
1 Linear Optimal Control Problem 389
2 SEM Discretization 390
3 Iteration and Discretization Error Estimates 392
4 Numerical Results 394
5 Conclusions 394
References 396
Applications of High Order Methods to Vortex Instability Calculations 397
1 Introduction 397
2 Theory 399
2.1 The Basic Flows 399
2.2 The BiGlobal Eigenvalue Problem (EVP) 400
3 Results 401
3.1 Basic Flow 401
3.2 Instability Analyses 401
4 Conclusions and Outlook 403
References 404
Entropy Viscosity Method for High-Order Approximations of Conservation Laws 405
1 Introduction 405
2 The Entropy Viscosity Method 406
3 2D Burgers (Fourier) 408
4 KPP Rotating Wave (SEM) 409
5 2D Euler System (Fourier) 411
References 412
High-Order Accurate Numerical Solution of Incompressible Slip Flow and Heat Transfer in Microchannels 413
1 Introduction 413
2 Problem Formulation 414
3 Wall Boundary Conditions 415
4 Numerical Procedure 415
5 Numerical Results and Discussion 416
6 Concluding Remarks 420
References 421
Spectral Methods for Time-Dependent Variable-Coefficient PDE Based on Block Gaussian Quadrature 422
1 Introduction 422
2 Krylov Subspace Spectral Methods 423
3 Implementation 426
4 Application to Maxwell's Equations 426
5 Numerical Results 428
5.1 Parabolic Problems 428
5.2 Maxwell's Equations 429
6 Summary and Future Work 431
References 431
The Spectral Element Method Used to Assess the Quality of a Global C1 Map 433
1 Introduction 433
2 Methods 434
3 Regularity 439
References 440
Stabilization of the Spectral-Element Method in Turbulent Flow Simulations 441
1 Introduction 441
2 Equations and Discretization 442
3 Stabilization of Turbulent Flow Simulations 443
4 Analysis of Model Problems 443
4.1 1D: Stabilization of the Burgers' Equation 443
4.2 2D: Recovery of Skew-Symmetry for the SEM Convection Operator in the Scalar Transport Equation 445
5 Application to the Navier–Stokes Equations 446
5.1 3D: Subcritical K-type Transition Simulations 447
5.2 3D: Fully Turbulent Channel Flow Simulationsat Re = 590 448
6 Conclusions 449
References 449
The Spectral-Element and Pseudo-Spectral Methods: A Comparative Study 451
1 Introduction 451
2 Study Setup 452
3 Results 453
3.1 Part A: Efficiency 453
3.2 Part B: Accuracy in Transitional Flow Simulations 454
3.3 Part B: Accuracy in Turbulent Flow Simulations 455
4 Conclusions 457
References 458
Adaptive Spectral Filtering and Digital Total Variation Postprocessing for the DG Method on Triangular Grids: Application to the Euler Equations 460
1 Introduction 460
2 The Discontinuous Galerkin Scheme with Spectral Filtering 461
3 The Digital Total Variation Filter 462
4 Numerical Experiments 463
References 467
BDDC and FETI-DP Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity 469
1 Introduction 469
2 Almost Incompressible Elasticity and Spectral Elements 469
3 The BDDC Algorithm 471
4 Numerical Results in the Plane 474
References 475
A Two-Dimensional DG-SEM Approach to Investigate Resonance Frequencies and Sound Radiation of Woodwind Instruments 477
1 Introduction 477
2 Discontinuous Galerkin Method for the Euler Equations 479
2.1 Conservation Equations 479
2.2 Numerical Scheme 479
3 The Influence of the Vocal Tract on the Recorder 480
3.1 Problem Description 480
3.2 Influence of the Vocal Tract 481
4 Sound Radiation of the Bassoon 482
5 Conclusions 484
References 484
Spectral Properties of Discontinuous Galerkin Space Operators on Curved Meshes 485
1 Introduction 485
2 Method 486
2.1 Discontinuous Galerkin Method 486
2.2 Stability Analysis 487
3 Results 488
3.1 Qualitative Results in 2D 489
3.2 Dependence on the Local Jacobian in 1D 490
3.3 Estimation Based on Integration Matrices in 1D 491
4 Conclusions 491
References 492
Post-Processing of Marginally Resolved Spectral Element Data 493
1 Introduction 493
2 Numerical Test Problems 494
2.1 An Analytical Example 494
2.2 Turbulent Channel Flow 494
3 Interface Averaging 495
4 Improved Interface Treatment 497
4.1 Polynomial Interpolation 497
4.2 Filtering 499
5 Conclusions 499
References 500
Editorial Policy 501
Lecture Notes in Computational Science and Engineering 503
Monographs in Computational Science and Engineering 507
Texts in Computational Science and Engineering 507