Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations (eBook)

Foreword 6
Preface 8
Contents 13
1 Classical Continuous Models and Their Analysis 18
1.1 The Basic Equations 18
1.1.1 The Acoustics Equation 19
1.1.2 Maxwell's Equations 19
1.1.3 The Linear Elastodynamics System 23
1.1.4 Boundary Conditions 25
1.2 Functional Issues 27
1.2.1 Some Functional Spaces 27
1.2.2 Variational Formulations 33
1.2.3 Energy Identities 37
1.2.4 Well-Posedness Results of Waves Equations 39
1.3 Plane Wave Solutions 47
1.3.1 A General Solution of the Homogeneous Wave Equation 47
1.3.2 Application to Maxwell's Equations 49
1.3.3 The 2D Case 51
1.3.4 Application to the Isotropic Linear Elastodynamics System 52
References 53
2 Definition of Different Types of Finite Elements 55
2.1 1D Mass-Lumping and Spectral Elements 55
2.1.1 A Complex Solution for a Simple Problem 55
2.1.2 Mass-Lumping 57
2.1.3 Spectral Elements 61
2.1.4 Nodal and Modal Elements 62
2.2 Quadrilaterals and Hexahedra 63
2.2.1 Higher-Dimensional Tensor Quadrature Rules 63
2.2.2 Tensor Unit Spectral Elements 64
2.2.3 Extension to Quadrilaterals and Hexahedra 66
2.3 Triangles and Tetrahedra 68
2.3.1 Spectral Triangles and Tetrahedra 68
2.3.2 Mass-Lumped Triangles and Tetrahedra 73
2.4 Purely 3D Elements 77
2.4.1 Wedges 78
2.4.2 Pyramids 78
2.5 Tetrahedral and Triangular Edge Elements 83
2.5.1 Mixed Formulation 83
2.5.2 A First Family 83
2.5.3 A Second Family 87
2.5.4 Tetrahedral Mass-Lumped Edge Elements 89
2.5.5 Triangular Mass-Lumped Edge Elements 92
2.6 Hexahedral and Quadrilateral Edge Elements 95
2.6.1 First Family 95
2.6.2 Second Family 98
2.7 H(div) Finite Elements 100
2.7.1 Tetrahedral and Triangular Elements 100
2.7.2 Hexahedral and Quadrilateral Elements 102
2.8 Other Mixed Elements 104
2.8.1 Pyramidal and Prismatic Edge Elements 104
2.8.2 Pyramidal and Prismatic H(div) Elements 106
References 107
3 Hexahedral and Quadrilateral Spectral Elements for Acoustic Waves 110
3.1 Second-Order Formulation of the Acoustics Equation 110
3.1.1 The Continuous and Approximate Problem 110
3.1.2 Discretization of the Integrals 111
3.2 First-Order Formulation of the Acoustics Equation 115
3.2.1 The Mixed Formulation 116
3.2.2 The Mass Matrices 117
3.2.3 The Stiffness Matrices 118
3.3 Comparison of the Methods 121
3.3.1 Matrix Formulation 121
3.3.2 A Theorem of Equivalence 122
3.3.3 Comparison of the Costs 124
3.4 Dispersion Relation 127
3.4.1 The Continuous Equation 128
3.4.2 A Didactic Case: The P1 Approximation 128
3.4.3 The Concept of Numerical Dispersion 131
3.4.4 P2 Approximation 132
3.4.5 P3 and Higher-Order Approximations 135
3.4.6 Extension to Higher Dimensions 141
3.5 Reflection-Transmission by a Discontinuous Interface 150
3.5.1 The Continuous Problem 150
3.5.2 FEM Approximation of the Heterogeneous Wave Equation 151
3.5.3 Taylor Expansion of the Wavenumber 152
3.5.4 Interface Between Two Elements 153
3.5.5 Interface at an Interior Point 154
3.5.6 Extension to Higher-Order Approximations 155
3.5.7 A Two-Layer Experiment 156
3.6 hp-a priori Error Estimates 160
3.6.1 Some Properties of Meshes 161
3.6.2 Some Interpolation Errors for Quadrilaterals and Hexahedra 163
3.6.3 hp-Estimation of Numerical Integration Errors 167
3.6.4 hp a priori Error Estimate for the Semi-discrete Approximation 174
3.7 The Linear Elastodynamics System 181
3.7.1 Second Order Formulation 181
3.7.2 First-Order Formulation 182
3.7.3 Comparison of the Two Approaches 185
References 188
4 Discontinuous Galerkin Methods 189
4.1 General Formulation for Linear Hyperbolic Problems 189
4.1.1 The Discontinuous Galerkin Formulation 189
4.1.2 Energy Identity 195
4.1.3 Application to Some Wave Equations 198
4.2 Approximation by Triangles and Tetrahedra 202
4.2.1 The Mass Integrals 203
4.2.2 The Stiffness Integrals 205
4.2.3 The Jump Terms 207
4.3 Approximation by Quadrilaterals and Hexahedra 211
4.3.1 The Mass Integrals 211
4.3.2 The Stiffness Integrals 212
4.3.3 The Jump Terms 213
4.3.4 Application to Wave Equations 215
4.4 Comparison of the DG Methods for Maxwell's Equations 221
4.4.1 Gauss or Gauss--Lobatto? 221
4.4.2 Tetrahedra with and Without Reconstruction of the Stiffness Matrix 224
4.4.3 Tetrahedra Versus Hexahedra 224
4.5 Plane Wave Analysis 232
4.5.1 The Eigenvalue Problem for the 1D Model 232
4.5.2 Numerical Dispersion and Dissipation 235
4.5.3 Extension to Higher Dimensions 238
4.6 Interior Penalty Discontinuous Galerkin Methods 240
4.6.1 General Formulation 240
4.6.2 Coercivity of the Discrete Operator 242
References 246
5 The Maxwell's System and Spurious Modes 247
5.1 A First Model and Its Approximation 247
5.1.1 The Continuous Model 247
5.1.2 The Approximate Model 248
5.1.3 The Discrete Mass Integral 249
5.2 A Second Model and Its Approximations 251
5.2.1 The Continuous Model 251
5.2.2 General Formulations of the Approximations 252
5.2.3 Approximation in H(Curl,?) 253
5.2.4 Approximation in [H1( ?)]3 254
5.2.5 Comparison of the Approximations 256
5.3 Suppressing Spurious Modes 259
5.3.1 Some Background About the Spurious Modes 259
5.3.2 Computation of the Eigenvalues of timestimes on a Cube 266
5.3.3 Discontinuous Galerkin Methods 269
5.3.4 The Second Family of Edge Elements 272
5.3.5 Continuous Elements 275
5.3.6 The Case of the First Family of Edge Elements 275
5.4 Error Estimates for DGM 277
5.4.1 The Discontinuous Galerkin Formulation 277
5.4.2 Choice of a Projector 278
5.4.3 hp-Projection Errors 281
5.4.4 Trace Lemmas 285
5.4.5 A Priori Error Estimates in Energy Norm 287
5.4.6 Extension to the Dissipative Case 294
References 296
6 Approximating Unbounded Domains 298
6.1 Absorbing Boundary Conditions (ABC) 299
6.1.1 Transparent Condition of the Wave Equation 299
6.1.2 Construction of ABC for the Wave Equation 300
6.1.3 Plane Wave Analysis 304
6.1.4 Finite Element Implementation 305
6.1.5 The Maxwell's System 309
6.2 Perfectly Matched Layers (PML) 310
6.2.1 Interpretation of the Method 310
6.2.2 The Acoustics System 313
6.2.3 The Maxwell's System 319
6.2.4 The Linear Elastodynamics System 321
6.2.5 Modified PML 323
References 325
7 Time Approximation 327
7.1 Schemes with a Constant Time-Step 327
7.1.1 Construction of the Schemes 328
7.1.2 Stability of the Schemes by Plane Wave Analysis 332
7.1.3 Stability of the Schemes by Energy Techniques 337
7.1.4 The Modified Equation and Unbounded Domains 339
7.1.5 A Remark About the Time Approximation of Dissipative DG Schemes 342
7.2 Local Time Stepping 346
7.2.1 Symplectic Schemes for Conservative Approximations 347
7.2.2 Scheme Based on a Lagrange Multiplier 352
7.2.3 An Explicit Conservative Scheme for Second-Order Wave Equations 358
References 365
8 Some Complex Models 367
8.1 The Linearized Euler Equations 367
8.1.1 Discontinuous Galerkin Approximation 368
8.1.2 H1-L2 Approximation 372
8.2 The Linear Cauchy--Poisson Problem 376
8.2.1 The Continuous Problem and Its Approximation 376
8.2.2 Unbounded Domains 379
8.3 Vibrating Thin Plates 384
8.3.1 The Continuous Models 385
8.3.2 Plane Wave Analysis 386
8.3.3 Mixed Spectral Element Approximation 390
References 392