Discontinuous Galerkin Method (eBook)

Preface 6
Contents 9
1 Introduction 15
1.1 DGM Versus Finite Volume and Finite Element Methods 16
1.2 A Short Historical Overview of the DGM 19
1.2.1 DGM for Hyperbolic and Singularly Perturbed Problems 19
1.2.2 DGM for Elliptic and Parabolic Problems 20
1.2.3 DGM for the Numerical Solution of Compressible Flow 22
1.2.4 Monographs Dealing with the DGM 23
1.3 Some Mathematical Concepts 23
1.3.1 Spaces of Continuous Functions 24
1.3.2 Lebesgue Spaces 25
1.3.3 Sobolev Spaces 26
1.3.4 Theorems on Traces and Embeddings 27
1.3.5 Bochner Spaces 30
1.3.6 Useful Theorems and Inequalities 33
Part I Analysis of the DiscontinuousGalerkin Method 38
2 DGM for Elliptic Problems 39
2.1 Model Problem 39
2.2 Abstract Numerical Method and Its Theoretical Analysis 40
2.3 Spaces of Discontinuous Functions 43
2.3.1 Partition of the Domain 43
2.3.2 Assumptions on Meshes 45
2.3.3 Broken Sobolev Spaces 47
2.4 DGM Based on a Primal Formulation 49
2.5 Basic Tools of the Theoretical Analysis of DGM 55
2.5.1 Multiplicative Trace Inequality 58
2.5.2 Inverse Inequality 60
2.5.3 Approximation Properties 61
2.6 Existence and Uniqueness of the Approximate Solution 63
2.6.1 The Choice of Penalty Weight ? 64
2.6.2 Continuity of Diffusion Bilinear Forms 65
2.6.3 Coercivity of Diffusion Bilinear Forms 71
2.7 Error Estimates 73
2.7.1 Estimates in the DG-Norm 73
2.7.2 Optimal L2(?)-Error Estimate 76
2.8 Baumann--Oden Method 80
2.9 Numerical Examples 84
2.9.1 Regular Solution 84
2.9.2 Singular Case 89
2.9.3 A Note on the L2(?)-Optimality of NIPG and IIPG 90
3 Methods Based on a Mixed Formulation 97
3.1 A General Mixed DG Method 97
3.1.1 Equivalent Formulations 99
3.1.2 Lifting Operators 100
3.2 Bassi--Rebay Methods 102
3.2.1 Mixed Formulation 102
3.2.2 Variational Formulation 103
3.2.3 Theoretical Analysis 107
3.3 Local Discontinuous Galerkin Method 116
3.3.1 Mixed Formulation 117
3.3.2 Variational Formulation 120
3.3.3 Theoretical Analysis 122
4 DGM for Convection-Diffusion Problems 128
4.1 Scalar Nonlinear Nonstationary Convection-Diffusion Equation 128
4.2 Discretization 131
4.3 Abstract Error Estimate 135
4.3.1 Consistency of the Convection Form in the Case of the Dirichlet Boundary Condition 136
4.3.2 Consistency of the Convective Form in the Case of Mixed Boundary Conditions 139
4.3.3 Error Estimates for the Method of Lines 145
4.4 Error Estimates in Terms of h 149
4.5 Optimal Linfty(0,T L2(?))-Error Estimate
4.6 Uniform Error Estimates with Respect to the Diffusion Coefficient 162
4.6.1 Continuous Problem 162
4.6.2 Discretization of the Problem 164
4.6.3 Error Estimates 168
4.7 Numerical Examples 177
5 Space-Time Discretization by Multistep Methods 181
5.1 Semi-implicit Backward Euler Time Discretization 181
5.1.1 Discretization of the Problem 182
5.1.2 Error Estimates 183
5.2 Backward Difference Formula for the Time Discretization 193
5.2.1 Discretization of the Problem 194
5.2.2 Theoretical Analysis 198
5.2.3 Error Estimates 208
5.2.4 Numerical Examples 231
6 Space-Time Discontinuous Galerkin Method 233
6.1 Space-Time DGM for a Heat Equation 233
6.1.1 Discretization of the Problem 234
6.1.2 Space-Time DG Discretization 236
6.1.3 Auxiliary Results 239
6.1.4 Space-Time Projection Operator 241
6.1.5 Abstract Error Estimate 251
6.1.6 Estimation of Projection Error in Terms of h and ? 255
6.1.7 Error Estimate in the DG-norm 263
6.1.8 Discrete Characteristic Function 265
6.1.9 Error Estimate in the Linfty(0,T L2(?))-norm
6.1.10 The Case of Identical Meshes on All Time Levels 274
6.1.11 Alternative Proof of Lemma6.12 274
6.2 Space-Time DGM for Nonlinear Convection-Diffusion Problems 277
6.2.1 Discretization of the Problem 278
6.2.2 Auxiliary Results 280
6.2.3 Abstract Error Estimate 289
6.2.4 Main Result 299
6.2.5 Numerical Examples 302
6.3 Extrapolated Space-Time Discontinuous Galerkin Method for Nonlinear ƒ 303
6.3.1 Discretization of the Problem 304
6.3.2 Auxiliary Results 307
6.3.3 Error Estimates 314
6.3.4 Numerical Examples 321
6.4 Uniform Error Estimates with Respect to the Diffusion Coefficient for the ST-DGM 327
6.4.1 Formulation of the Problem and Some Assumptions 329
6.4.2 Discretization of the Problem 329
6.4.3 Properties of the Discrete Problem 332
6.4.4 Abstract Error Estimate 333
6.4.5 Numerical Examples 342
7 Generalization of the DGM 346
7.1 hp-Discontinuous Galerkin Method 346
7.1.1 Formulation of a Model Problem 347
7.1.2 Discretization 347
7.1.3 Theoretical Analysis 351
7.1.4 Computational Performance of the hp-DGM 361
7.2 DGM on General Elements 370
7.2.1 Assumptions on the Domain Partition 371
7.2.2 Function Spaces 372
7.2.3 Approximate Solution 373
7.2.4 Auxiliary Results 374
7.2.5 Error Analysis 379
7.2.6 Numerical Examples 379
7.3 The Effect of Numerical Integration 383
7.3.1 Continuous Problem 383
7.3.2 Space Semidiscretization 384
7.3.3 Numerical Integration 385
7.3.4 Some Important Results 386
7.3.5 Truncation Error of Quadrature Formulae 388
7.3.6 Properties of the Convection Forms 392
7.3.7 The Effect of Numerical Integration in the Convection Form 396
7.3.8 Error Estimates for the Method of Lines with Numerical Integration 401
Part II Applications of the DiscontinuousGalerkin Method 407
8 Inviscid Compressible Flow 408
8.1 Formulation of the Inviscid Flow Problem 409
8.1.1 Governing Equations 409
8.1.2 Initial and Boundary Conditions 415
8.2 DG Space Semidiscretization 416
8.2.1 Notation 416
8.2.2 Discontinuous Galerkin Space Semidiscretization 418
8.3 Numerical Treatment of Boundary Conditions 420
8.3.1 Boundary Conditions on Impermeable Walls 420
8.3.2 Boundary Conditions on the Inlet and Outlet 423
8.4 Time Discretization 430
8.4.1 Backward Euler Method 431
8.4.2 Newton Method Based on the Jacobi Matrix 432
8.4.3 Newton-Like Method Based on the Flux Matrix 433
8.4.4 Realization of the Iterative Algorithm 439
8.4.5 Higher-Order Time Discretization 441
8.4.6 Choice of the Time Step 446
8.4.7 Structure of the Flux Matrix 448
8.4.8 Construction of the Basis in the Space Shp 450
8.4.9 Steady-State Solution 452
8.5 Shock Capturing 454
8.5.1 Jump Indicators 455
8.5.2 Artificial Viscosity Shock Capturing 456
8.5.3 Numerical Examples 458
8.6 Approximation of a Nonpolygonal Boundary 463
8.6.1 Curved Elements 463
8.6.2 DGM Over Curved Elements 465
8.6.3 Numerical Examples 470
8.7 Numerical Verification of the BDF-DGM 474
8.7.1 Inviscid Low Mach Number Flow 474
8.7.2 Low Mach Number Flow at Incompressible Limit 476
8.7.3 Isentropic Vortex Propagation 479
8.7.4 Supersonic Flow 481
9 Viscous Compressible Flow 483
9.1 Formulation of the Viscous Compressible Flow Problem 483
9.1.1 Governing Equations 483
9.1.2 Initial and Boundary Conditions 489
9.2 DG Space Semidiscretization 491
9.2.1 Notation 491
9.2.2 DG Space Semidiscretization of Viscous Terms 492
9.2.3 Semidiscrete Problem 497
9.3 Time Discretization 498
9.3.1 Time Discretization Schemes 498
9.3.2 Solution Strategy 499
9.4 Numerical Examples 503
9.4.1 Blasius Problem 504
9.4.2 Stationary Flow Around the NACA 0012 Profile 510
9.4.3 Unsteady Flow 516
9.4.4 Steady Versus Unsteady Flow 518
9.4.5 Viscous Shock-Vortex Interaction 520
10 Fluid-Structure Interaction 526
10.1 Formulation of Flow in a Time-Dependent Domain 526
10.1.1 Space Discretization of the Flow Problem 528
10.1.2 Time Discretization by the BDF Method 533
10.1.3 Space-Time DG Discretization 535
10.2 Fluid-Structure Interaction 536
10.2.1 Flow-Induced Airfoil Vibrations 536
10.2.2 Interaction of Compressible Flow and an Elastic Body 542
References 557
Index 570