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Domain Decomposition Methods in Science and Engineering XIX

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472 Seiten
Englisch
Springer Berlin Heidelbergerschienen am27.10.20102011
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.mehr
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KlappentextThese 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/Kritik

Inhaltsverzeichnis
1;Preface;5
2;Contents;10
3;Contributors;15
4;Part I Plenary Presentations;23
4.1;Domain Decomposition and hp-Adaptive Finite Elements;24
4.1.1;1 Introduction;24
4.1.2;2 A Posteriori Error Estimate;25
4.1.3;3 Basis Functions;26
4.1.4;4 Parallel Adaptive Algorithm;27
4.1.5;5 DD Solver;28
4.1.6;6 Numerical Results;31
4.1.7;Bibliography;34
4.2;Domain Decomposition Methods for Electromagnetic Wave Propagation Problems in Heterogeneous Media and Complex Domains;35
4.2.1;1 Introduction;35
4.2.2;2 Continuous Problem;36
4.2.3;3 A Family of Schwarz DD Algorithms;37
4.2.4;4 Discretization by a High Order DG Method;38
4.2.4.1;4.1 Discretization of the Monodomain Problem;38
4.2.4.2;4.2 Discretization of the DD Algorithm;39
4.2.4.2.1;DG Formulation of the Multi-Domain Problem;39
4.2.4.2.2;Formulation of an Interface System;40
4.2.5;5 Numerical Results;41
4.2.5.1;5.1 The 2D Case;41
4.2.5.2;5.2 The 3D Case;43
4.2.6;6 Ongoing and Future Work;44
4.2.7;Bibliography;45
4.3;N--N Solvers for a DG Discretization for Geometrically Nonconforming Substructures and Discontinuous Coefficients ;47
4.3.1;1 Summary;47
4.3.2;2 Introduction;47
4.3.3;3 Differential and Discrete Problems;49
4.3.3.1;3.1 Differential Problem;49
4.3.3.2;3.2 Discrete Problem;49
4.3.3.3;3.3 Schur Complement Problem;50
4.3.4;4 Notation and the Interface Condition;53
4.3.5;5 Additive Preconditioners;55
4.3.5.1;5.1 Local Problems;55
4.3.5.2;5.2 Coarse Problems;56
4.3.5.3;5.3 Condition Number Estimate for Tas,I;56
4.3.6;6 Final Remarks;57
4.3.7;Bibliography;57
4.4;On Adaptive-Multilevel BDDC;59
4.4.1;1 Introduction;59
4.4.2;2 Abstract BDDC for a Model Problem;60
4.4.2.1;2.1 Multilevel BDDC;61
4.4.3;3 Indicator of the Condition Number Bound;63
4.4.4;4 Optimal Coarse Degrees of Freedom;64
4.4.5;5 Adaptive-Multilevel BDDC in 2D;65
4.4.6;6 Numerical Examples and Conclusion;66
4.4.7;Bibliography;70
4.5;Interpolation Based Local Postprocessing for Adaptive Finite Element Approximations in Electronic Structure Calculations;71
4.5.1;1 Introduction;71
4.5.2;2 Interpolation Based Finite Element Postprocessing;73
4.5.2.1;2.1 Finite Element Discretizations;74
4.5.2.2;2.2 Interpolation Based Local Postprocessing;75
4.5.2.3;2.3 Quantum Harmonic Oscillator;75
4.5.3;3 Applications to Electronic Structure Calculations;76
4.5.3.1;3.1 Linearization of Kohn--Sham Equation;76
4.5.3.2;3.2 Experiments;77
4.5.3.2.1;Benzene;78
4.5.3.2.2;Fullerene;79
4.5.4;4 Concluding Remarks;80
4.5.5;Bibliography;80
4.6;A New a Posteriori Error Estimate for Adaptive Finite Element Methods;82
4.6.1;1 Introduction;82
4.6.2;2 A Posteriori Error Estimate;83
4.6.3;3 Numerical Validation and Applications;89
4.6.4;Bibliography;92
4.7;Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries;94
4.7.1;1 Introduction;94
4.7.2;2 The Continuous OSWR Algorithm;95
4.7.3;3 Numerical Results;100
4.7.4;4 Conclusions;103
4.7.5;Bibliography;105
4.8;Convergence Behaviour of Dirichlet--Neumann and Robin Methods for a Nonlinear Transmission Problem;106
4.8.1;1 Introduction;106
4.8.2;2 Transmission Problem with Jumping Nonlinearities;108
4.8.3;3 Nonlinear Dirichlet--Neumann and Robin Methods;109
4.8.3.1;3.1 The Methods and Their Steklov--Poincaré Formulations;109
4.8.3.2;3.2 Convergence Results;110
4.8.4;4 Parameter Studies for the Dirichlet--Neumann Method;111
4.8.5;5 Parameter Studies for the Robin Method;114
4.8.6;Bibliography;117
5;Part II Minisymposia;118
5.1;Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains;119
5.1.1;1 Optimal Interface Conditions;119
5.1.2;2 Notation and Assumptions;120
5.1.3;3 Construction of the Method;120
5.1.4;4 Sparsity Pattern;122
5.1.5;5 Numerical Examples;124
5.1.6;6 Conclusion;125
5.1.7;Bibliography;126
5.2;Optimized Schwarz Methods for Domains with an Arbitrary Interface;127
5.2.1;1 Introduction;127
5.2.2;2 First-Order Boundary Condition;128
5.2.3;3 Higher-Order Boundary Condition;130
5.2.4;Bibliography;133
5.3;Can the Discretization Modify the Performance of Schwarz Methods?;135
5.3.1;1 Introduction;135
5.3.2;2 The Cauchy--Riemann Equations;135
5.3.3;3 The Positive Definite Helmholtz Equation;140
5.3.4;4 Conclusions;141
5.3.5;Bibliography;141
5.4;The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator;143
5.4.1;1 Introduction;143
5.4.2;2 Model Problem;144
5.4.3;3 The Pole Condition;145
5.4.4;4 Error Estimate;147
5.4.5;Bibliography;149
5.5;Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation;151
5.5.1;1 Introduction;151
5.5.2;2 Local Problem and Time Discontinuous Galerkin;152
5.5.3;3 The Optimized Schwarz Waveform Relaxation Algorithm Discretized in Time with Different Subdomain Grids;153
5.5.4;4 Numerical Results;156
5.5.5;5 Conclusions;158
5.5.6;Bibliography;158
5.6;Two-Level Methods for Blood Flow Simulation;159
5.6.1;1 Introduction;159
5.6.2;2 Mathematical Model and Discretization;159
5.6.3;3 Two-Level Newton and Schwarz Methods;161
5.6.4;4 Numerical Results;163
5.6.5;5 Conclusion;166
5.6.6;Bibliography;166
5.7;Newton-Krylov-Schwarz Method for a Spherical Shallow Water Model;167
5.7.1;1 Introduction;167
5.7.2;2 Governing Equations;167
5.7.3;3 Discretizations;168
5.7.4;4 Nonlinear Solver;169
5.7.5;5 Numerical Results;170
5.7.6;Bibliography;172
5.8;A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation;174
5.8.1;1 Introduction;174
5.8.2;2 A Description of the ASPJD Algorithm;175
5.8.3;3 A PETSc-Based ASPJD Polynomial Eigensolver;177
5.8.4;4 Numerical Results;178
5.8.5;Bibliography;180
5.9;Two-Level Multiplicative Domain Decomposition Algorithm for Recovering the Lamé Coefficient in Biological Tissues;182
5.9.1;1 Introduction;182
5.9.2;2 Recovering the Lamé Coefficient in Biological Tissues;182
5.9.3;3 Lagrange-Newton-Krylov-Schwarz Algorithm;184
5.9.4;4 Numerical Results and Discussion;186
5.9.5;5 Concluding Remarks;188
5.9.6;Bibliography;189
5.10;Robust Preconditioner for H(curl) Interface Problems;190
5.10.1;1 Introduction;190
5.10.2;2 Regular Decomposition;191
5.10.3;3 Auxiliary Space Preconditioners;194
5.10.4;4 Conclusions;196
5.10.5;Bibliography;196
5.11;Mixed Multiscale Finite Element Analysis for Wave Equations Using Global Information;198
5.11.1;1 Introduction;198
5.11.2;2 Preliminaries;199
5.11.3;3 Mixed MsFEM Analysis;200
5.11.3.1;3.1 Mixed MsFEM Formulation;200
5.11.3.2;3.2 A Priori Error Estimates for Continuous Time;202
5.11.3.3;3.3 A Priori Error Estimate for Discrete Time;204
5.11.4;4 Conclusions;205
5.11.5;Bibliography;205
5.12;A Domain Decomposition Preconditioner for Multiscale High-Contrast Problems;206
5.12.1;1 Summary;206
5.12.2;2 Introduction;206
5.12.3;3 Problem Setting and Domain Decomposition Framework;207
5.12.4;4 Coarse-Space-Completing Eigenvalue Problem and Stability Estimates;209
5.12.5;5 Numerical Results;211
5.12.6;Bibliography;213
5.13;Weighted Poincaré Inequalities and Applications in Domain Decomposition;214
5.13.1;1 Introduction;214
5.13.2;2 Weighted Poincaré Inequalities;215
5.13.3;3 Explicit Dependence on Geometrical Parameters;217
5.13.4;Bibliography;220
5.14;Technical Tools for Boundary Layers and Applications to Heterogeneous Coefficients;222
5.14.1;1 Summary;222
5.14.2;2 Introduction and Assumptions;222
5.14.3;3 Technical Tools for Layers;224
5.14.3.1;3.1 Technical Tools for DDMs;225
5.14.4;4 Dual-Primal Formulation;226
5.14.5;5 FETI-DP Preconditioner;228
5.14.6;Bibliography;229
5.15;Coarse Spaces over the Ages;230
5.15.1;1 Introduction;230
5.15.2;2 Local Nullspace and Bounded Energy Conditions;230
5.15.3;3 Some Early Domain Decomposition Methods;232
5.15.4;4 Balancing Domain Decomposition (BDD) and FETI;233
5.15.5;5 BDDC and FETI-DP;234
5.15.6;6 Adaptive Methods by Enriching the Coarse Space;235
5.15.7;Bibliography;235
5.16;FETI-DP for Stokes-Mortar-Darcy Systems;238
5.16.1;1 Introduction and Problem Setting;238
5.16.2;2 Weak Formulation;239
5.16.3;3 Discretization and Decomposition;240
5.16.4;4 Dual Formulation;242
5.16.4.1;4.1 Dirichlet Preconditioner;243
5.16.5;5 Numerical Results;244
5.16.6;Bibliography;245
5.17;Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids;246
5.17.1;1 Introduction;246
5.17.2;2 Constraint Decomposition Methods;247
5.17.3;3 A Constraint Decomposition on Bisection Grids;248
5.17.4;4 Numerical Experiments;252
5.17.5;Bibliography;252
5.18;How Close to the Fully Viscous Solution Can One Get with Inviscid Approximations in Subregions ?;254
5.18.1;1 Introduction;254
5.18.2;2 Model Problem;255
5.18.3;3 Factorization of the Differential Operator;256
5.18.4;4 Optimal Coupling Conditions and Approximations;257
5.18.5;5 Numerical Asymptotic Study;258
5.18.6;6 Conclusions;260
5.18.7;Bibliography;260
5.19;Schwarz Waveform Relaxation Algorithms with Nonlinear Transmission Conditions for Reaction-Diffusion Equations;262
5.19.1;1 Introduction;262
5.19.2;2 Problem Description;263
5.19.3;3 The Schwarz Waveform Relaxation Algorithm;263
5.19.3.1;3.1 Non-overlapping Algorithms of Order Zero and Two;264
5.19.3.2;3.2 Well-Posedness and Convergence;264
5.19.4;4 Discretization;265
5.19.4.1;4.1 Nonlinear Transmission Conditions;265
5.19.4.2;4.2 Implementation of the Iterative Algorithm;266
5.19.5;5 Numerical Results;267
5.19.5.1;5.1 A Simple Model in Geological CO2 Storage Modeling;268
5.19.6;Bibliography;269
5.20;Recent Advances in Schwarz Waveform Moving Mesh Methods -- A New Moving Subdomain Method;270
5.20.1;1 Introduction;270
5.20.2;2 Moving Meshes;270
5.20.3;3 Domain Decomposition Strategies;272
5.20.3.1;3.1 SWR in Physical Co-ordinates -- Existing Methods;273
5.20.3.2;3.2 SWR in Computational Co-ordinates -- A New Approach;274
5.20.4;4 Numerical Results and Comments;275
5.20.5;Bibliography;277
5.21;Optimized Schwarz Waveform Relaxation Methods: A Large Scale Numerical Study;278
5.21.1;1 Introduction;278
5.21.2;2 Optimized Schwarz Waveform Relaxation;278
5.21.3;3 Theoretical Results;279
5.21.4;4 Numerical Experiments;282
5.21.5;5 Conclusions;284
5.21.6;Bibliography;285
5.22;Optimized Schwarz Methods for Maxwell's Equations with Non-zero Electric Conductivity;286
5.22.1;1 Introduction;286
5.22.2;2 Schwarz Methods for Maxwell's Equations;286
5.22.3;3 Analysis for Non-zero Electric Conductivity;288
5.22.4;4 Numerical Results;291
5.22.5;5 Conclusion;293
5.22.6;Bibliography;293
5.23;Robust Boundary Element Domain Decomposition Solvers in Acoustics;294
5.23.1;1 Introduction;294
5.23.2;2 Formulation of the Domain Decomposition Approach;294
5.23.3;3 Construction of Preconditioners;297
5.23.3.1;3.1 Local Preconditioners;297
5.23.3.2;3.2 Global Preconditioners;298
5.23.4;4 Numerical Examples;299
5.23.4.1;4.1 Local Preconditioners;299
5.23.4.2;4.2 Global Preconditioners;300
5.23.5;Bibliography;301
5.24;A Newton Based Fluid--Structure Interaction Solver with Algebraic Multigrid Methods on Hybrid Meshes;302
5.24.1;1 Problem Setting of the Fluid--Structure Interaction;302
5.24.1.1;1.1 Geometrical Description;302
5.24.1.2;1.2 The Physical Model;303
5.24.1.3;1.3 Reformulation of the Model;304
5.24.1.4;1.4 Time Semi-Discretized Weak Formulations;305
5.24.1.4.1;Time Semi-discretized Structure Weak Formulation;305
5.24.1.4.2;Time Semi-discretized Fluid Weak Formulation;305
5.24.1.4.3;The Variational Form of the Interface Equation;306
5.24.2;2 Newton's Method for the Interface Equation;307
5.24.3;3 Finite Element Discretization on Hybrid Meshes;307
5.24.4;4 AMG for the Structure and the Fluid Sub-problems;307
5.24.5;5 Numerical Results;308
5.24.6;Bibliography;309
5.25;Coupled FE/BE Formulations for the Fluid--Structure Interaction;310
5.25.1;1 Introduction;310
5.25.2;2 Integral Equations and Variational Formulations;311
5.25.3;3 Symmetric Coupling of Finite and Boundary Elements;312
5.25.4;4 Nonsymmetric Finite and Boundary Element Coupling;314
5.25.4.1;4.1 A Second Kind Boundary Integral Equation Approach;314
5.25.4.2;4.2 A First Kind Boundary Integral Equation Approach;315
5.25.5;5 Conclusions;316
5.25.6;Bibliography;317
5.26;Domain Decomposition Solvers for Frequency-Domain Finite Element Equations;318
5.26.1;1 Introduction;318
5.26.2;2 Frequency-Domain Finite Element Equations;319
5.26.3;3 Domain Decomposition Solver;321
5.26.4;4 A Symmetric and Indefinite Reformulation;322
5.26.5;5 Conclusions, Outlook, and Acknowledgments;324
5.26.6;Bibliography;324
5.27;Deriving the X-Z Identity from Auxiliary Space Method;326
5.27.1;1 Iterative Methods;326
5.27.2;2 Auxiliary Space Method;327
5.27.3;3 Auxiliary Spaces of Product Type;329
5.27.4;4 Method of Subspace Correction;331
5.27.5;Bibliography;333
5.28;A Near-Optimal Hierarchical Estimate Based Adaptive Finite Element Method for Obstacle Problems;334
5.28.1;1 Introduction;334
5.28.2;2 A Near-Optimal Hierarchical Error Estimate;335
5.28.3;3 An Adaptive Finite Element Method;337
5.28.4;4 Numerical Experiments;338
5.28.5;Bibliography;340
5.29;Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems;342
5.29.1;1 Introduction;342
5.29.2;2 A Parallel Preconditioner for the H(grad) System;343
5.29.2.1;2.1 A Parallel AMG Preconditioner;344
5.29.2.2;2.2 Numerical Experiments;346
5.29.3;3 A Parallel Preconditioner for the H(curl) Problem;347
5.29.3.1;3.1 A Parallel Preconditioner for (5);347
5.29.3.2;3.2 Numerical Results;348
5.29.4;Bibliography;349
6;Part III Contributed Presentations;350
6.1;A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization;351
6.1.1;1 Introduction;351
6.1.2;2 Interior Penalty Discontinuous Galerkin Methods;352
6.1.3;3 Space Decomposition;354
6.1.3.1;3.1 Matrix Representation of the DG Bilinear Forms;355
6.1.4;4 A Uniformly Convergent Iterative Method;356
6.1.5;5 Numerical Results;356
6.1.6;Bibliography;358
6.2;A Study of Prolongation OperatorsBetween Non-nested Meshes;359
6.2.1;1 Introduction;359
6.2.2;2 Multilevel Preconditioners Based on Non-nested Meshes;360
6.2.3;3 Looking for Suitable Prolongation Operators;362
6.2.4;4 Numerical Results;364
6.2.5;Bibliography;366
6.3;A Parallel Schwarz Method for Multiple Scattering Problems;367
6.3.1;1 Introduction;367
6.3.2;2 Exterior Helmholtz Problem and Schwarz Method;368
6.3.2.1;2.1 Domain Decomposition;368
6.3.2.2;2.2 A Parallel Schwarz Method;368
6.3.3;3 Multiple DtN Operator;369
6.3.4;4 How to Solve Problem (2);371
6.3.5;5 Proof of Theorem 1;371
6.3.6;6 Concluding Remarks;373
6.3.7;Bibliography;374
6.4;Numerical Method for Antenna Radiation Problem by FDTD Method with PML;375
6.4.1;1 FDTD Method and PML;375
6.4.2;2 Basic Formulation of Antenna Problem;377
6.4.3;3 Application to MRI Problem;379
6.4.4;4 Summary and Future Problems;380
6.4.5;Bibliography;381
6.5;On Domain Decomposition Algorithms for Contact Problems with Tresca Friction;382
6.5.1;1 Introduction;382
6.5.2;2 Contact Problems with Tresca Friction;382
6.5.3;3 Algorithms and the Implementation;383
6.5.4;4 Numerical Experiments;386
6.5.5;5 Conclusions and Comments;388
6.5.6;Bibliography;388
6.6;Numerical Solution of Linear Elliptic Problems with Robin Boundary Conditions by a Least-Squares/Fictitious Domain Method;390
6.6.1;1 Introduction;390
6.6.2;2 Formulation of the Boundary Value Problem;390
6.6.3;3 A Least-Squares/Fictitious Domain Method for the Solution of Problem (1), (2);391
6.6.3.1;3.1 A Fictitious Domain Formulation of Problem (1), (2);391
6.6.3.2;3.2 A Least-Squares Formulation of Problem (7);392
6.6.4;4 On the Conjugate Gradient Solution of the Least-Squares Problem (8);392
6.6.5;5 On the Finite Element Implementation of the Least-Squares/ Fictitious Domain Methodology;394
6.6.5.1;5.1 Generalities;394
6.6.5.2;5.2 Finite Element Approximation of the Least-Squares Problem (8);394
6.6.6;6 Numerical Experiments;395
6.6.7;Bibliography;397
6.7;An Uzawa Domain Decomposition Method for Stokes Problem;398
6.7.1;1 Introduction;398
6.7.2;2 Model Problem;398
6.7.3;3 Uzawa Domain Decomposition for Stokes Problem;399
6.7.3.1;3.1 Lagrangian Formulation and Dual Problem;400
6.7.3.2;3.2 Sensitivity Analysis;401
6.7.3.3;3.3 Uzawa Conjugate Gradient Domain Decomposition Algorithm;402
6.7.4;4 Numerical Experiments;403
6.7.5;5 Conclusion;405
6.7.6;Bibliography;405
6.8;A Domain Decomposition Method Combining a Boundary Element Method with a Meshless Local Petrov-Galerkin Method;406
6.8.1;1 Introduction;406
6.8.2;2 A DDM Combining BEM with the MLPG Method;407
6.8.3;3 A Dynamic Relaxation Parameter;410
6.8.4;4 Numerical Examples;410
6.8.5;5 Conclusions;412
6.8.6;Bibliography;412
6.9;A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term in Three Dimensions;414
6.9.1;1 Introduction;414
6.9.2;2 Dual Iterative Substructuring with a Penalty Term;415
6.9.3;3 Estimate of Condition Number;418
6.9.4;4 Computational Issues;419
6.9.5;Bibliography;421
6.10;Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients;422
6.10.1;1 Summary;422
6.10.2;2 Introduction;422
6.10.3;3 Notation and Building Tools;423
6.10.4;4 Multigrid Method;426
6.10.5;5 Multilevel Additive Preconditioner (BPX);426
6.10.6;6 Condition Number Bounds;427
6.10.7;7 Numerical Experiments;427
6.10.8;Bibliography;429
6.11;A FETI-DP Formation for the Stokes Problem Without Primal Pressure Components;430
6.11.1;1 Introduction;430
6.11.2;2 FETI-DP Formulation;431
6.11.2.1;2.1 Model Problem;431
6.11.2.2;2.2 FETI-DP Formulation Without Primal Pressure Components;432
6.11.3;3 Analysis of a Bound of Condition Number;435
6.11.3.1;3.1 Lower Bound;435
6.11.3.2;3.2 Upper Bound;436
6.11.4;Bibliography;437
6.12;Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations;438
6.12.1;1 Introduction;438
6.12.2;2 Systems of Semi-linear Reaction Diffusion Equations;439
6.12.3;3 Schwarz Waveform Relaxation Algorithm;440
6.12.4;4 Numerical Results;442
6.12.4.1;4.1 Belousov-Zhabotinsky Equations;442
6.12.4.2;4.2 FitzHugh-Nagumo Equations;443
6.12.4.3;4.3 Lotka-Volterra Equations;443
6.12.5;5 Conclusions;445
6.12.6;Bibliography;445
6.13;A Sparse QS-Decomposition for Large Sparse Linear System of Equations;446
6.13.1;1 Introduction;446
6.13.2;2 A Quasi-Orthogonal Vector Sequence;447
6.13.3;3 Layered Group Orthogonalization;448
6.13.3.1;3.1 Algorithm (LGO);448
6.13.3.2;3.2 Matrix representation of LGO;449
6.13.4;4 LGO Solver and Numerical Experiments;449
6.13.5;5 A Nested Direct Domain Decomposition Idea;451
6.13.6;Bibliography;453
6.14;Is Additive Schwarz with Harmonic Extension Just Lions' Method in Disguise?;454
6.14.1;1 The Methods of Lions, AS, RAS and ASH;454
6.14.2;2 Assumptions and the Main Result;456
6.14.3;3 Proof of the Main Result;457
6.14.4;4 Convergence Rate;459
6.14.5;5 Conclusions;461
6.14.6;Bibliography;461
6.15;Domain Decomposition Methods for a Complementarity Problem;462
6.15.1;1 Introduction;462
6.15.2;2 Semismooth Function Approaches for Complementarity Problems;463
6.15.2.1;2.1 Semismooth Newton Methods;463
6.15.2.2;2.2 Schwarz Preconditioner;465
6.15.3;3 Numerical Experiments;465
6.15.3.1;3.1 One-Level Results;466
6.15.3.2;3.2 Two-Level Results;466
6.15.4;4 Some Final Remarks;467
6.15.5;Bibliography;469
6.16;A Posteriori Error Estimates for Semilinear Boundary Control Problems;470
6.16.1;1 Introduction;470
6.16.2;2 Finite Elements for Boundary Control Problems;471
6.16.3;3 A Posteriori Error Estimates;473
6.16.4;Bibliography;477
6.17;Lecture Notes in Computational Science and Engineering;480mehr

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