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Domain Decomposition Methods in Science and Engineering XVIII

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376 Seiten
Englisch
Springer Berlin Heidelbergerschienen am01.09.20092009
th This volume contains a selection of 41 refereed papers presented at the 18 International Conference of Domain Decomposition Methods hosted by the School of ComputerScience and Engineering(CSE) of the Hebrew Universityof Jerusalem, Israel, January 12-17, 2008. 1 Background of the Conference Series The International Conference on Domain Decomposition Methods has been held in twelve countries throughout Asia, Europe, the Middle East, and North America, beginning in Paris in 1987. Originally held annually, it is now spaced at roughly 18-month intervals. A complete list of past meetings appears below. The principal technical content of the conference has always been mathematical, but the principal motivation has been to make ef cient use of distributed memory computers for complex applications arising in science and engineering. The leading 15 such computers, at the 'petascale' characterized by 10 oating point operations per second of processing power and as many Bytes of application-addressablem- ory, now marshal more than 200,000 independentprocessor cores, and systems with many millions of cores are expected soon. There is essentially no alternative to - main decomposition as a stratagem for parallelization at such scales. Contributions from mathematicians, computerscientists, engineers,and scientists are together n- essary in addressing the challenge of scale, and all are important to this conference.mehr
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Klappentextth This volume contains a selection of 41 refereed papers presented at the 18 International Conference of Domain Decomposition Methods hosted by the School of ComputerScience and Engineering(CSE) of the Hebrew Universityof Jerusalem, Israel, January 12-17, 2008. 1 Background of the Conference Series The International Conference on Domain Decomposition Methods has been held in twelve countries throughout Asia, Europe, the Middle East, and North America, beginning in Paris in 1987. Originally held annually, it is now spaced at roughly 18-month intervals. A complete list of past meetings appears below. The principal technical content of the conference has always been mathematical, but the principal motivation has been to make ef cient use of distributed memory computers for complex applications arising in science and engineering. The leading 15 such computers, at the 'petascale' characterized by 10 oating point operations per second of processing power and as many Bytes of application-addressablem- ory, now marshal more than 200,000 independentprocessor cores, and systems with many millions of cores are expected soon. There is essentially no alternative to - main decomposition as a stratagem for parallelization at such scales. Contributions from mathematicians, computerscientists, engineers,and scientists are together n- essary in addressing the challenge of scale, and all are important to this conference.

Inhalt/Kritik

Inhaltsverzeichnis
1;Preface;5
2;Contents;13
3;Part I Plenary Presentations;17
3.1;A Domain Decomposition Approach for Calculating the GraphCorresponding to a Fibrous Geometry;18
3.1.1;1 Introduction;18
3.1.2;2 Preliminaries;19
3.1.3;3 Statement of the Problem;20
3.1.4;4 A Divide and Conquer Algorithm;20
3.1.4.1;4.1 Numerical Complexity of Algorithm 1;21
3.1.5;5 Numerical Results and Conclusions;23
3.1.6;References;29
3.2;Adaptive Multilevel Interior-Point Methods in PDE ConstrainedOptimization;30
3.2.1;1 Introduction;30
3.2.2;2 Optimal Design of Processes and Systems;31
3.2.3;3 Adaptive Multilevel Primal-Dual Interior Point Methods;32
3.2.4;4 Numerical Results;36
3.2.5;References;39
3.3;Numerical Homogeneisation Technique with Domain DecompositionBased a-posteriori Error Estimates;42
3.3.1;1 Introduction;42
3.3.2;2 Mechanical Problem;43
3.3.3;3 Numerical Homogeneisation;44
3.3.4;4 Error Control;45
3.3.4.1;4.1 Motivation and Reference Local Solution;45
3.3.4.2;4.2 Adjoint Equation;47
3.3.4.3;4.3 Explicit a Posteriori Error Estimate;48
3.3.4.4;4.4 Numerical Solution of the Adjoint Problem;49
3.3.5;5 Numerical Results;49
3.3.6;6 Conclusions;51
3.3.7;References;51
3.4;Multiscale Methods for Multiphase Flow in Porous Media;53
3.4.1;1 Introduction;53
3.4.2;2 A Framework for Discussing Multiscale Methods;54
3.4.3;3 A Model Problem for Multiphase Flow;55
3.4.4;4 Some Upscaling Methods;56
3.4.4.1;4.1 Permeability Upscaling;56
3.4.4.2;4.2 Saturation Upscaling;58
3.4.4.3;4.3 Vertically Integrated Models;59
3.4.5;5 Multiscale Numerical Methods;60
3.4.5.1;5.1 The Variational Multiscale Method;60
3.4.5.2;5.2 A VMS Approach for the Implicit Time-Discretized Pressure Equation;61
3.4.6;6 Conclusions;62
3.4.7;References;62
3.5;Mixed Plane Wave Discontinuous Galerkin Methods;65
3.5.1;1 Introduction;65
3.5.2;2 Mixed Discontinuous Galerkin Approach;66
3.5.3;3 Convergence Analysis of the Mixed PWDG Method;68
3.5.4;4 Conclusion;75
3.5.5;References;75
3.6;Numerical Zoom and the Schwarz Algorithm;77
3.6.1;1 Introduction;77
3.6.1.1;Chimera;77
3.6.1.2;Hilbert Space Decomposition Method;78
3.6.1.3;Harmonic Patch Iterator;79
3.6.1.4;One Way Schwarz;80
3.6.2;2 Convergence of Schwarz' Algorithm on Arbitrary Non-Matching Meshes;81
3.6.2.1;Numerical Tests;84
3.6.3;3 Numerical Comparison of the Methods;85
3.6.4;4 Conclusion;87
3.6.5;References;87
3.7;BDDC for Nonsymmetric Positive Definite and Symmetric IndefiniteProblems;88
3.7.1;1 Introduction;88
3.7.2;2 Finite Element Discretization;89
3.7.2.1;2.1 Nonsymmetric, Positive Definite Problems;89
3.7.2.2;2.2 Symmetric, Indefinite Problems;90
3.7.3;3 The BDDC Preconditioners;91
3.7.4;4 Convergence Rate Analysis;92
3.7.4.1;4.1 Nonsymmetric, Positive Cases;93
3.7.4.2;4.2 Symmetric, Indefinite Cases;94
3.7.5;5 Numerical Experiments;95
3.7.5.1;5.1 Nonsymmetric, Positive Definite Cases;95
3.7.5.2;5.2 Symmetric and Indefinite Cases;95
3.7.6;References;98
3.8;Accomodating Irregular Subdomains in Domain Decomposition Theory;100
3.8.1;1 Introduction;100
3.8.2;2 A Poincaré Inequality, John and Jones Domains;102
3.8.3;3 FETI-DP and BDDC Algorithms;104
3.8.4;4 An Overlapping Schwarz Method;107
3.8.5;5 Almost Incompressible Elasticity;109
3.8.6;References;110
3.9;Auxiliary Space Preconditioners for Mixed Finite Element Methods;112
3.9.1;1 Introduction;112
3.9.2;2 HX-Preconditioner for H(div) Systems;113
3.9.3;3 Application to Mixed Method;116
3.9.4;4 Numerical Results;118
3.9.4.1;4.1 Constant Coefficients;118
3.9.4.2;4.2 Variable Coefficients;118
3.9.4.3;4.3 Augmented Lagrangian Iterations;120
3.9.5;5 Conclusions;121
3.9.6;References;121
4;Part II Minisymposia;123
4.1;A Multilevel Domain Decomposition Solver Suited to NonsmoothMechanical Problems;124
4.1.1;1 Introduction;124
4.1.2;2 A Multiscale Description;124
4.1.3;3 Preliminary: Linear Elastic Case;125
4.1.4;4 Nonsmooth Case: a Tensegrity Grid;128
4.1.5;5 Conclusions;130
4.1.6;References;130
4.2;A FETI-2LM Method for Non-Matching Grids;132
4.2.1;1 Introduction;132
4.2.2;2 FETI-2LM method;132
4.2.2.1;2.1 Discrete Approach;132
4.2.2.2;2.2 Optimal Interface Operator;134
4.2.3;3 Mortar Method;135
4.2.4;4 A FETI-2LM Method for Non-conforming Interfaces;136
4.2.5;5 Localization of Non-conforming Interface Matching Conditions;137
4.2.6;6 Conclusion;138
4.2.7;References;139
4.3;Truncated Nonsmooth Newton Multigrid Methods for ConvexMinimization Problems;140
4.3.1;1 Introduction;140
4.3.2;2 A Nonsmooth Newton Method;141
4.3.3;3 Multigrid;143
4.3.4;4 Example I: Two-Body Contact in Linear Elasticity;144
4.3.5;5 Example II: The Allen-Cahn Equation;145
4.3.6;References;147
4.4;A Recursive Trust-Region Method for Non-Convex ConstrainedMinimization;148
4.4.1;1 Introduction;148
4.4.2;2 The Multilevel Setting;149
4.4.3;3 Recursive Trust-Region Methods;150
4.4.3.1;3.1 Convergence to First-Order Critical Points;151
4.4.4;4 Numerical Example;154
4.4.5;References;154
4.5;A Robin Domain Decomposition Algorithm for Contact Problems:Convergence Results;156
4.5.1;1 Introduction;156
4.5.2;2 Weak Formulation of the Continuous Problem;156
4.5.3;3 The Domain Decomposition Algorithm;158
4.5.4;4 Convergence;159
4.5.5;5 Numerical Experiments;161
4.5.6;References;162
4.6;Patch Smoothers for Saddle Point Problems with Applicationsto PDE-Constrained Optimization Problems;164
4.6.1;1 Introduction;164
4.6.2;2 An Optimal Control Problem;165
4.6.3;3 The Multigrid Method;167
4.6.3.1;3.1 The Patch Smoother;168
4.6.4;4 Numerical Experiments;169
4.6.5;References;171
4.7;A Domain Decomposition Preconditioner of Neumann-NeumannType for the Stokes Equations;172
4.7.1;1 Introduction;172
4.7.2;2 DDM for the Stokes Equations;173
4.7.2.1;2.1 Stokes Equations;173
4.7.2.2;2.2 A New Algorithm for the Stokes Equations;173
4.7.3;3 Numerical Results;174
4.7.4;References;178
4.8;Non-overlapping Domain Decomposition for the Richards Equationvia Superposition Operators;180
4.8.1;1 Introduction;180
4.8.2;2 Weak Forms of the Domain Decomposition Problems;182
4.8.3;3 Kirchhoff Transformation as a Superposition Operator;183
4.8.4;4 Equivalence of the Weak Formulations;186
4.8.5;References;187
4.9;Convergence Behavior of a Two-Level Optimized Schwarz Preconditioner;188
4.9.1;1 Introduction;188
4.9.2;2 Domain Decomposition Preconditioners;189
4.9.2.1;2.1 One-Level Preconditioners;189
4.9.2.2;2.2 Interpretation of the Algebraic Condition;190
4.9.2.3;2.3 Two-Level Preconditioners;191
4.9.3;3 Numerical Results;192
4.9.3.1;3.1 Dependence on h;192
4.9.3.2;3.2 Dependence on H, with Generous Overlap;193
4.9.3.3;3.3 A Weak Scalability Test;193
4.9.4;4 Best Robin Parameter;193
4.9.5;References;195
4.10;An Algorithm for Non-Matching Grid Projections with Linear Complexity;196
4.10.1;1 Introduction;196
4.10.2;2 Towards an Optimal Algorithm;196
4.10.3;3 The Algorithm for Computing the Intersection;198
4.10.4;4 The Projection Algorithm with Linear Complexity;199
4.10.5;5 Numerical Experiments;201
4.10.6;6 Conclusions;201
4.10.7;References;202
4.11;A Maximum Principle for L2-Trace Norms withan Application to Optimized Schwarz Methods;204
4.11.1;1 Introduction;204
4.11.2;2 The Maximum Principle for L2-Trace Norms;205
4.11.2.1;2.1 Preliminaries on the Domain and the Interfaces;205
4.11.2.2;2.2 (,a,c)-Harmonicity;206
4.11.2.3;2.3 -Relative Uniformity;207
4.11.2.4;2.4 Maximum Principle for L2-Trace Norms;208
4.11.3;3 Applications to Schwarz Methods;210
4.11.4;References;211
4.12;An Extended Mathematical Framework for Barrier Methodsin Function Space;212
4.12.1;1 Convex State Constrained Optimal Control;212
4.12.1.1;1.1 Linear Equality Constraints;213
4.12.1.2;1.2 Inequality Constraints and Convex Functionals;213
4.12.1.3;1.3 Example: A class of Elliptic PDEs;214
4.12.2;2 The Homotopy Path and its Properties;215
4.12.3;References;219
4.13;Optimized Schwarz Preconditioning for SEM BasedMagnetohydrodynamics;220
4.13.1;1 Introduction;220
4.13.2;2 Governing Equations and Discretization;221
4.13.3;3 From Classical to Optimized Schwarz;223
4.13.4;4 Discretization of the Optimized Schwarz;224
4.13.5;5 Numerical Experiments;226
4.13.6;References;227
4.14;Nonlinear Overlapping Domain Decomposition Methods;228
4.14.1;1 Introduction;228
4.14.2;2 Newton-Krylov-Schwarz Algorithms;230
4.14.3;3 Classical Schwarz Alternating Algorithms;231
4.14.4;4 Nonlinear Additive Schwarz Preconditioned Inexact Newton Algorithms;231
4.14.5;5 Nonlinear Elimination Algorithms;232
4.14.6;6 Nonlinear Restricted Additive Schwarz Algorithms;233
4.14.7;7 Concluding Remarks;234
4.14.8;References;235
4.15;Optimized Schwarz Waveform Relaxation: Roots, Blossoms and Fruits;236
4.15.1;1 Introduction: Parallel Processing of Evolution Problems;236
4.15.2;2 Roots: Waveform Relaxation for ODEs;237
4.15.3;3 Blossoms: Classical Schwarz Waveform Relaxation for Parabolic Equations;238
4.15.4;4 Fruits: Optimized Schwarz Waveform Relaxationfor Parabolic Equations;239
4.15.5;5 Other Fruits: Optimized Schwarz Waveform Relaxationfor Other Types of PDEs;240
4.15.6;6 New Blossoms: Space-Time Coupling and Refinements;241
4.15.7;References;242
4.16;Optimized Schwarz Methods;244
4.16.1;1 Introduction: Original Schwarz Method (1870);244
4.16.1.1;1.1 Towards Faster Methods: Two Families of Methods;245
4.16.2;2 Modified Schwarz Method;246
4.16.2.1;2.1 Generalized Schwarz Methods;246
4.16.2.2;2.2 Optimal Interface Conditions;247
4.16.3;3 Conclusion and Open Problems;249
4.16.4;References;249
4.17;The Development of Coarse Spaces for DomainDecomposition Algorithms;252
4.17.1;1 Introduction;252
4.17.2;2 Early Two-Level Domain Decomposition Methods;253
4.17.3;3 Additional Comments;255
4.17.4;4 Other Iterative Substructuring Methods;255
4.17.5;5 FETI-DP and BDDC;256
4.17.6;6 Additional Roles for Coarse Spaces;257
4.17.7;References;257
5;Part III Contributed Presentations;260
5.1;Distributed Decomposition Over Hyperspherical Domains;261
5.1.1;1 Global Optimization for Semiconductor LithographyMask Design;261
5.1.1.1;1.1 Convex Partitions of the Feasible Domain;262
5.1.2;2 Partitions of n-Space;263
5.1.3;3 Incomplete Search Heuristics;263
5.1.4;4 Hypersphere Decomposition;264
5.1.4.1;4.1 Previous Work;264
5.1.4.2;4.2 A Memory-Efficient Tree Storage Scheme for Equal AreaHypersphere Regions;265
5.1.4.3;4.3 Parallel Decomposition;266
5.1.5;5 Ongoing Work;267
5.1.6;References;267
5.2;Domain Decomposition Preconditioning for Discontinuous GalerkinApproximations of Convection-Diffusion Problems;269
5.2.1;1 Introduction;269
5.2.2;2 Statement of the Problem and its DG Approximation;269
5.2.3;3 Nonoverlapping Schwarz Methods;271
5.2.4;4 The Issue of Convergence;272
5.2.5;5 Numerical Experiments;273
5.2.6;References;276
5.3;Linearly Implicit Domain Decomposition Methods for NonlinearTime-Dependent Reaction-Diffusion Problems;277
5.3.1;1 Introduction;277
5.3.2;2 Spatial Discretization;278
5.3.3;3 Time Integration;279
5.3.4;4 Numerical Results;282
5.3.5;References;284
5.4;NKS for Fully Coupled Fluid-Structure Interaction with Application;285
5.4.1;1 Introduction;285
5.4.2;2 Governing Equations;285
5.4.3;3 Spatial Discretization;287
5.4.4;4 Temporal Discretization;288
5.4.5;5 Solving the Nonlinear System;289
5.4.6;6 Numerical Results;290
5.4.7;7 Conclusion;291
5.4.8;References;292
5.5;Weak Information Transfer between Non-Matching Warped Interfaces;293
5.5.1;1 Introduction;293
5.5.2;2 Discrete Information Transfer;294
5.5.3;3 The Discrete Coupling Operator;295
5.5.4;4 Numerical Results;299
5.5.5;References;300
5.6;Computational Tool for a Mini-Windmill Study with SOFT;301
5.6.1;1 Introduction and Motivation;301
5.6.2;2 Flow Solver;303
5.6.3;3 FSI;304
5.6.4;4 Parallel Computing Scenario and Conclusion;306
5.6.5;References;307
5.7;On Preconditioners for Generalized Saddle Point Problemswith an Indefinite Block;308
5.7.1;1 Introduction;308
5.7.2;2 General Assumptions;309
5.7.3;3 Block Diagonal Preconditioner;311
5.7.4;4 Block Upper Triangular Preconditioner;311
5.7.5;5 Lower Block Triangular Preconditioner;312
5.7.6;6 Numerical Experiments;313
5.7.7;7 Conclusions;314
5.7.8;References;314
5.8;Lower Bounds for Eigenvalues of Elliptic Operators by OverlappingDomain Decomposition;316
5.8.1;1 Introduction;316
5.8.2;2 Description of the Method;317
5.8.3;3 Two Simple Examples;321
5.8.4;References;322
5.9;From the Boundary Element Domain Decomposition Methods to LocalTrefftz Finite Element Methods on Polyhedral Meshes;324
5.9.1;1 Introduction;324
5.9.2;2 The Potential Equation;325
5.9.3;3 The Helmholtz Equation;328
5.9.4;4 The Maxwell Equations;328
5.9.5;5 Conclusions;330
5.9.6;References;330
5.10;An Additive Neumann-Neumann Method for Mortar Finite Elementfor 4th Order Problems;332
5.10.1;1 Introduction;332
5.10.2;2 Discrete Problem;333
5.10.3;3 Neumann-Neumann Method;335
5.10.3.1;3.1 Local Subspaces;335
5.10.3.2;3.2 Coarse Space;336
5.10.4;References;338
5.11;A Numerically Efficient Scheme for Elastic Immersed Boundaries;340
5.11.1;1 Introduction;340
5.11.2;2 Discretization of the IBM;340
5.11.3;3 Volume Conservation Method Based on Constrained Optimization;342
5.11.4;4 Application of the IBM and Conclusion;345
5.11.5;References;346
5.12;A Domain Decomposition Method Based on Augmented Lagrangianwith a Penalty Term;348
5.12.1;1 Introduction;348
5.12.2;2 Saddle-Point Formulation;349
5.12.3;3 Iterative Substructuring Method;350
5.12.4;4 Computational Issues and Numerical Results;352
5.12.4.1;4.1 Computational Issues;352
5.12.4.2;4.2 Numerical Results;353
5.12.5;5 Conclusions;354
5.12.6;References;355
5.13;Parallelization of a Constrained Three-Dimensional Maxwell Solver;356
5.13.1;1 Introduction;356
5.13.2;2 Constrained Wave Equation Formulation;357
5.13.3;3 Variational Formulations;357
5.13.4;4 Space and Time Discretization;359
5.13.5;5 Solution of the Doubly Constrained System;361
5.13.6;6 Numerical Application;362
5.13.7;7 Conclusion;363
5.13.8;References;363
5.14;A Discovery Algorithm for the Algebraic Construction of OptimizedSchwarz Preconditioners;364
5.14.1;1 Introduction;364
5.14.2;2 Discovery Algorithm;365
5.14.2.1;2.1 Interface Detection;366
5.14.2.2;2.2 Extraction of Physical and Discretization Parameters;366
5.14.2.3;2.3 Construction of the Optimized Transmission Condition;367
5.14.3;3 Numerical Experiments;369
5.14.4;References;371
5.15;On the Convergence of Optimized Schwarz Methodsby way of Matrix Analysis;372
5.15.1;1 Introduction;372
5.15.1.1;1.1 Model Problem and Notation;373
5.15.2;2 Convergence of OSM;375
5.15.3;3 Numerical Experiments;377
5.15.4;References;378mehr

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