Domain Decomposition Methods in Science and Engineering XIX (eBook)

Yunqing Huang, Ralf Kornhuber, Olof Widlund, Jinchao Xu (Herausgeber)

eBook Download: PDF

2010
XXIV, 472 Seiten
Springer Berlin (Verlag)
978-3-642-11304-8 (ISBN)

Infos

These are the proceedings of the 19th international conference on domain decomposition methods in science and engineering. Domain decomposition methods are iterative methods for solving the often very large linear or nonlinear systems of algebraic equations that arise in various problems in mathematics, computational science, engineering and industry. They are designed for massively parallel computers and take the memory hierarchy of such systems into account. This is essential for approaching peak floating point performance. There is an increasingly well-developed theory which is having a direct impact on the development and improvement of these algorithms.

The editors are all well-known researchers in this research field.

Inhalt

Preface 5
Contents 10
Contributors 15
Part I Plenary Presentations 23
Domain Decomposition and hp-Adaptive Finite Elements 24
1 Introduction 24
2 A Posteriori Error Estimate 25
3 Basis Functions 26
4 Parallel Adaptive Algorithm 27
5 DD Solver 28
6 Numerical Results 31
Bibliography 34
Domain Decomposition Methods for Electromagnetic Wave Propagation Problems in Heterogeneous Media and Complex Domains 35
1 Introduction 35
2 Continuous Problem 36
3 A Family of Schwarz DD Algorithms 37
4 Discretization by a High Order DG Method 38
4.1 Discretization of the Monodomain Problem 38
4.2 Discretization of the DD Algorithm 39
DG Formulation of the Multi-Domain Problem 39
Formulation of an Interface System 40
5 Numerical Results 41
5.1 The 2D Case 41
5.2 The 3D Case 43
6 Ongoing and Future Work 44
Bibliography 45
N--N Solvers for a DG Discretization for Geometrically Nonconforming Substructures and Discontinuous Coefficients 47
1 Summary 47
2 Introduction 47
3 Differential and Discrete Problems 49
3.1 Differential Problem 49
3.2 Discrete Problem 49
3.3 Schur Complement Problem 50
4 Notation and the Interface Condition 53
5 Additive Preconditioners 55
5.1 Local Problems 55
5.2 Coarse Problems 56
5.3 Condition Number Estimate for Tas,I 56
6 Final Remarks 57
Bibliography 57
On Adaptive-Multilevel BDDC 59
1 Introduction 59
2 Abstract BDDC for a Model Problem 60
2.1 Multilevel BDDC 61
3 Indicator of the Condition Number Bound 63
4 Optimal Coarse Degrees of Freedom 64
5 Adaptive-Multilevel BDDC in 2D 65
6 Numerical Examples and Conclusion 66
Bibliography 70
Interpolation Based Local Postprocessing for Adaptive Finite Element Approximations in Electronic Structure Calculations 71
1 Introduction 71
2 Interpolation Based Finite Element Postprocessing 73
2.1 Finite Element Discretizations 74
2.2 Interpolation Based Local Postprocessing 75
2.3 Quantum Harmonic Oscillator 75
3 Applications to Electronic Structure Calculations 76
3.1 Linearization of Kohn--Sham Equation 76
3.2 Experiments 77
Benzene 78
Fullerene 79
4 Concluding Remarks 80
Bibliography 80
A New a Posteriori Error Estimate for Adaptive Finite Element Methods 82
1 Introduction 82
2 A Posteriori Error Estimate 83
3 Numerical Validation and Applications 89
Bibliography 92
Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries 94
1 Introduction 94
2 The Continuous OSWR Algorithm 95
3 Numerical Results 100
4 Conclusions 103
Bibliography 105
Convergence Behaviour of Dirichlet--Neumann and Robin Methods for a Nonlinear Transmission Problem 106
1 Introduction 106
2 Transmission Problem with Jumping Nonlinearities 108
3 Nonlinear Dirichlet--Neumann and Robin Methods 109
3.1 The Methods and Their Steklov--Poincaré Formulations 109
3.2 Convergence Results 110
4 Parameter Studies for the Dirichlet--Neumann Method 111
5 Parameter Studies for the Robin Method 114
Bibliography 117
Part II Minisymposia 118
Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains 119
1 Optimal Interface Conditions 119
2 Notation and Assumptions 120
3 Construction of the Method 120
4 Sparsity Pattern 122
5 Numerical Examples 124
6 Conclusion 125
Bibliography 126
Optimized Schwarz Methods for Domains with an Arbitrary Interface 127
1 Introduction 127
2 First-Order Boundary Condition 128
3 Higher-Order Boundary Condition 130
Bibliography 133
Can the Discretization Modify the Performance of Schwarz Methods? 135
1 Introduction 135
2 The Cauchy--Riemann Equations 135
3 The Positive Definite Helmholtz Equation 140
4 Conclusions 141
Bibliography 141
The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator 143
1 Introduction 143
2 Model Problem 144
3 The Pole Condition 145
4 Error Estimate 147
Bibliography 149
Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation 151
1 Introduction 151
2 Local Problem and Time Discontinuous Galerkin 152
3 The Optimized Schwarz Waveform Relaxation Algorithm Discretized in Time with Different Subdomain Grids 153
4 Numerical Results 156
5 Conclusions 158
Bibliography 158
Two-Level Methods for Blood Flow Simulation 159
1 Introduction 159
2 Mathematical Model and Discretization 159
3 Two-Level Newton and Schwarz Methods 161
4 Numerical Results 163
5 Conclusion 166
Bibliography 166
Newton-Krylov-Schwarz Method for a Spherical Shallow Water Model 167
1 Introduction 167
2 Governing Equations 167
3 Discretizations 168
4 Nonlinear Solver 169
5 Numerical Results 170
Bibliography 172
A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation 174
1 Introduction 174
2 A Description of the ASPJD Algorithm 175
3 A PETSc-Based ASPJD Polynomial Eigensolver 177
4 Numerical Results 178
Bibliography 180
Two-Level Multiplicative Domain Decomposition Algorithm for Recovering the Lamé Coefficient in Biological Tissues 182
1 Introduction 182
2 Recovering the Lamé Coefficient in Biological Tissues 182
3 Lagrange-Newton-Krylov-Schwarz Algorithm 184
4 Numerical Results and Discussion 186
5 Concluding Remarks 188
Bibliography 189
Robust Preconditioner for H(curl) Interface Problems 190
1 Introduction 190
2 Regular Decomposition 191
3 Auxiliary Space Preconditioners 194
4 Conclusions 196
Bibliography 196
Mixed Multiscale Finite Element Analysis for Wave Equations Using Global Information 198
1 Introduction 198
2 Preliminaries 199
3 Mixed MsFEM Analysis 200
3.1 Mixed MsFEM Formulation 200
3.2 A Priori Error Estimates for Continuous Time 202
3.3 A Priori Error Estimate for Discrete Time 204
4 Conclusions 205
Bibliography 205
A Domain Decomposition Preconditioner for Multiscale High-Contrast Problems 206
1 Summary 206
2 Introduction 206
3 Problem Setting and Domain Decomposition Framework 207
4 Coarse-Space-Completing Eigenvalue Problem and Stability Estimates 209
5 Numerical Results 211
Bibliography 213
Weighted Poincaré Inequalities and Applications in Domain Decomposition 214
1 Introduction 214
2 Weighted Poincaré Inequalities 215
3 Explicit Dependence on Geometrical Parameters 217
Bibliography 220
Technical Tools for Boundary Layers and Applications to Heterogeneous Coefficients 222
1 Summary 222
2 Introduction and Assumptions 222
3 Technical Tools for Layers 224
3.1 Technical Tools for DDMs 225
4 Dual-Primal Formulation 226
5 FETI-DP Preconditioner 228
Bibliography 229
Coarse Spaces over the Ages 230
1 Introduction 230
2 Local Nullspace and Bounded Energy Conditions 230
3 Some Early Domain Decomposition Methods 232
4 Balancing Domain Decomposition (BDD) and FETI 233
5 BDDC and FETI-DP 234
6 Adaptive Methods by Enriching the Coarse Space 235
Bibliography 235
FETI-DP for Stokes-Mortar-Darcy Systems 238
1 Introduction and Problem Setting 238
2 Weak Formulation 239
3 Discretization and Decomposition 240
4 Dual Formulation 242
4.1 Dirichlet Preconditioner 243
5 Numerical Results 244
Bibliography 245
Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids 246
1 Introduction 246
2 Constraint Decomposition Methods 247
3 A Constraint Decomposition on Bisection Grids 248
4 Numerical Experiments 252
Bibliography 252
How Close to the Fully Viscous Solution Can One Get with Inviscid Approximations in Subregions ? 254
1 Introduction 254
2 Model Problem 255
3 Factorization of the Differential Operator 256
4 Optimal Coupling Conditions and Approximations 257
5 Numerical Asymptotic Study 258
6 Conclusions 260
Bibliography 260
Schwarz Waveform Relaxation Algorithms with Nonlinear Transmission Conditions for Reaction-Diffusion Equations 262
1 Introduction 262
2 Problem Description 263
3 The Schwarz Waveform Relaxation Algorithm 263
3.1 Non-overlapping Algorithms of Order Zero and Two 264
3.2 Well-Posedness and Convergence 264
4 Discretization 265
4.1 Nonlinear Transmission Conditions 265
4.2 Implementation of the Iterative Algorithm 266
5 Numerical Results 267
5.1 A Simple Model in Geological CO2 Storage Modeling 268
Bibliography 269
Recent Advances in Schwarz Waveform Moving Mesh Methods -- A New Moving Subdomain Method 270
1 Introduction 270
2 Moving Meshes 270
3 Domain Decomposition Strategies 272
3.1 SWR in Physical Co-ordinates -- Existing Methods 273
3.2 SWR in Computational Co-ordinates -- A New Approach 274
4 Numerical Results and Comments 275
Bibliography 277
Optimized Schwarz Waveform Relaxation Methods: A Large Scale Numerical Study 278
1 Introduction 278
2 Optimized Schwarz Waveform Relaxation 278
3 Theoretical Results 279
4 Numerical Experiments 282
5 Conclusions 284
Bibliography 285
Optimized Schwarz Methods for Maxwell's Equations with Non-zero Electric Conductivity 286
1 Introduction 286
2 Schwarz Methods for Maxwell's Equations 286
3 Analysis for Non-zero Electric Conductivity 288
4 Numerical Results 291
5 Conclusion 293
Bibliography 293
Robust Boundary Element Domain Decomposition Solvers in Acoustics 294
1 Introduction 294
2 Formulation of the Domain Decomposition Approach 294
3 Construction of Preconditioners 297
3.1 Local Preconditioners 297
3.2 Global Preconditioners 298
4 Numerical Examples 299
4.1 Local Preconditioners 299
4.2 Global Preconditioners 300
Bibliography 301
A Newton Based Fluid--Structure Interaction Solver with Algebraic Multigrid Methods on Hybrid Meshes 302
1 Problem Setting of the Fluid--Structure Interaction 302
1.1 Geometrical Description 302
1.2 The Physical Model 303
1.3 Reformulation of the Model 304
1.4 Time Semi-Discretized Weak Formulations 305
Time Semi-discretized Structure Weak Formulation 305
Time Semi-discretized Fluid Weak Formulation 305
The Variational Form of the Interface Equation 306
2 Newton's Method for the Interface Equation 307
3 Finite Element Discretization on Hybrid Meshes 307
4 AMG for the Structure and the Fluid Sub-problems 307
5 Numerical Results 308
Bibliography 309
Coupled FE/BE Formulations for the Fluid--Structure Interaction 310
1 Introduction 310
2 Integral Equations and Variational Formulations 311
3 Symmetric Coupling of Finite and Boundary Elements 312
4 Nonsymmetric Finite and Boundary Element Coupling 314
4.1 A Second Kind Boundary Integral Equation Approach 314
4.2 A First Kind Boundary Integral Equation Approach 315
5 Conclusions 316
Bibliography 317
Domain Decomposition Solvers for Frequency-Domain Finite Element Equations 318
1 Introduction 318
2 Frequency-Domain Finite Element Equations 319
3 Domain Decomposition Solver 321
4 A Symmetric and Indefinite Reformulation 322
5 Conclusions, Outlook, and Acknowledgments 324
Bibliography 324
Deriving the X-Z Identity from Auxiliary Space Method 326
1 Iterative Methods 326
2 Auxiliary Space Method 327
3 Auxiliary Spaces of Product Type 329
4 Method of Subspace Correction 331
Bibliography 333
A Near-Optimal Hierarchical Estimate Based Adaptive Finite Element Method for Obstacle Problems 334
1 Introduction 334
2 A Near-Optimal Hierarchical Error Estimate 335
3 An Adaptive Finite Element Method 337
4 Numerical Experiments 338
Bibliography 340
Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems 342
1 Introduction 342
2 A Parallel Preconditioner for the H(grad) System 343
2.1 A Parallel AMG Preconditioner 344
2.2 Numerical Experiments 346
3 A Parallel Preconditioner for the H(curl) Problem 347
3.1 A Parallel Preconditioner for (5) 347
3.2 Numerical Results 348
Bibliography 349
Part III Contributed Presentations 350
A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization 351
1 Introduction 351
2 Interior Penalty Discontinuous Galerkin Methods 352
3 Space Decomposition 354
3.1 Matrix Representation of the DG Bilinear Forms 355
4 A Uniformly Convergent Iterative Method 356
5 Numerical Results 356
Bibliography 358
A Study of Prolongation OperatorsBetween Non-nested Meshes 359
1 Introduction 359
2 Multilevel Preconditioners Based on Non-nested Meshes 360
3 Looking for Suitable Prolongation Operators 362
4 Numerical Results 364
Bibliography 366
A Parallel Schwarz Method for Multiple Scattering Problems 367
1 Introduction 367
2 Exterior Helmholtz Problem and Schwarz Method 368
2.1 Domain Decomposition 368
2.2 A Parallel Schwarz Method 368
3 Multiple DtN Operator 369
4 How to Solve Problem (2) 371
5 Proof of Theorem 1 371
6 Concluding Remarks 373
Bibliography 374
Numerical Method for Antenna Radiation Problem by FDTD Method with PML 375
1 FDTD Method and PML 375
2 Basic Formulation of Antenna Problem 377
3 Application to MRI Problem 379
4 Summary and Future Problems 380
Bibliography 381
On Domain Decomposition Algorithms for Contact Problems with Tresca Friction 382
1 Introduction 382
2 Contact Problems with Tresca Friction 382
3 Algorithms and the Implementation 383
4 Numerical Experiments 386
5 Conclusions and Comments 388
Bibliography 388
Numerical Solution of Linear Elliptic Problems with Robin Boundary Conditions by a Least-Squares/Fictitious Domain Method 390
1 Introduction 390
2 Formulation of the Boundary Value Problem 390
3 A Least-Squares/Fictitious Domain Method for the Solution of Problem (1), (2) 391
3.1 A Fictitious Domain Formulation of Problem (1), (2) 391
3.2 A Least-Squares Formulation of Problem (7) 392
4 On the Conjugate Gradient Solution of the Least-Squares Problem (8) 392
5 On the Finite Element Implementation of the Least-Squares/ Fictitious Domain Methodology 394
5.1 Generalities 394
5.2 Finite Element Approximation of the Least-Squares Problem (8) 394
6 Numerical Experiments 395
Bibliography 397
An Uzawa Domain Decomposition Method for Stokes Problem 398
1 Introduction 398
2 Model Problem 398
3 Uzawa Domain Decomposition for Stokes Problem 399
3.1 Lagrangian Formulation and Dual Problem 400
3.2 Sensitivity Analysis 401
3.3 Uzawa Conjugate Gradient Domain Decomposition Algorithm 402
4 Numerical Experiments 403
5 Conclusion 405
Bibliography 405
A Domain Decomposition Method Combining a Boundary Element Method with a Meshless Local Petrov-Galerkin Method 406
1 Introduction 406
2 A DDM Combining BEM with the MLPG Method 407
3 A Dynamic Relaxation Parameter 410
4 Numerical Examples 410
5 Conclusions 412
Bibliography 412
A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term in Three Dimensions 414
1 Introduction 414
2 Dual Iterative Substructuring with a Penalty Term 415
3 Estimate of Condition Number 418
4 Computational Issues 419
Bibliography 421
Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients 422
1 Summary 422
2 Introduction 422
3 Notation and Building Tools 423
4 Multigrid Method 426
5 Multilevel Additive Preconditioner (BPX) 426
6 Condition Number Bounds 427
7 Numerical Experiments 427
Bibliography 429
A FETI-DP Formation for the Stokes Problem Without Primal Pressure Components 430
1 Introduction 430
2 FETI-DP Formulation 431
2.1 Model Problem 431
2.2 FETI-DP Formulation Without Primal Pressure Components 432
3 Analysis of a Bound of Condition Number 435
3.1 Lower Bound 435
3.2 Upper Bound 436
Bibliography 437
Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations 438
1 Introduction 438
2 Systems of Semi-linear Reaction Diffusion Equations 439
3 Schwarz Waveform Relaxation Algorithm 440
4 Numerical Results 442
4.1 Belousov-Zhabotinsky Equations 442
4.2 FitzHugh-Nagumo Equations 443
4.3 Lotka-Volterra Equations 443
5 Conclusions 445
Bibliography 445
A Sparse QS-Decomposition for Large Sparse Linear System of Equations 446
1 Introduction 446
2 A Quasi-Orthogonal Vector Sequence 447
3 Layered Group Orthogonalization 448
3.1 Algorithm (LGO) 448
3.2 Matrix representation of LGO 449
4 LGO Solver and Numerical Experiments 449
5 A Nested Direct Domain Decomposition Idea 451
Bibliography 453
Is Additive Schwarz with Harmonic Extension Just Lions' Method in Disguise? 454
1 The Methods of Lions, AS, RAS and ASH 454
2 Assumptions and the Main Result 456
3 Proof of the Main Result 457
4 Convergence Rate 459
5 Conclusions 461
Bibliography 461
Domain Decomposition Methods for a Complementarity Problem 462
1 Introduction 462
2 Semismooth Function Approaches for Complementarity Problems 463
2.1 Semismooth Newton Methods 463
2.2 Schwarz Preconditioner 465
3 Numerical Experiments 465
3.1 One-Level Results 466
3.2 Two-Level Results 466
4 Some Final Remarks 467
Bibliography 469
A Posteriori Error Estimates for Semilinear Boundary Control Problems 470
1 Introduction 470
2 Finite Elements for Boundary Control Problems 471
3 A Posteriori Error Estimates 473
Bibliography 477
Lecture Notes in Computational Science and Engineering 480

Produkt-Details

• Erscheint lt. Verlag: 27.10.2010
• Reihe/Serie: Lecture Notes in Computational Science and Engineering
• Reihe/Serie: Lecture Notes in Computational Science and Engineering
• Zusatzinfo: XXIV, 472 p. 107 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: domain decomposition • finite elements • Parallel Computing • preconditioned conjugate gradients
• ISBN-10: 3-642-11304-4 / 3642113044
• ISBN-13: 978-3-642-11304-8 / 9783642113048