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Domain Decomposition Methods - Algorithms and Theory : Springer Series in Computational Mathematics - Andrea Toselli

Domain Decomposition Methods - Algorithms and Theory

Springer Series in Computational Mathematics

Hardcover Published: 1st November 2004
ISBN: 9783540206965
Number Of Pages: 450

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The purpose of this text is to offer a comprehensive and self-contained presentation of some of the most successful and popular domain decomposition preconditioners for finite and spectral element approximations of partial differential equations. Strong emphasis is placed on both algorithmic and mathematical aspects. Some important methods such FETI and balancing Neumann-Neumann methods and algorithms for spectral element methods, not treated previously in any monograph, are covered in detail.

From the reviews of the first edition:

"This book unifies the results from a number of papers by the authors and their coworkers over the past two decades, and complements them by new insights and some background. The distinguishing feature of this book is a comprehensive and rigorous treatment of convergence bounds based on the theory of infinite elements. ... The bibliography is quite complete for the fields covered ... . The book belongs on the desk of all specialists involved in domain decomposition and substructuring ... ." (Jan Mandel, Zentralblatt MATH, Vol. 1069, 2005)

Introductionp. 1
Basic Ideas of Domain Decompositionp. 1
Matrix and Vector Representationsp. 2
Nonoverlapping Methodsp. 5
An Equation for u¿: the Schur Complement Systemp. 5
An Equation for the Fluxp. 6
The Dirichlet-Neumann Algorithmp. 8
The Neumann-Neumann Algorithmp. 10
A Dirichlet-Dirichlet Algorithm or a FETI Methodp. 12
The Case of Many Subdomainsp. 15
The Schwarz Alternating Methodp. 21
Description of the Methodp. 21
The Schwarz Alternating Method as a Richardson Methodp. 22
Block Jacobi Preconditionersp. 24
Some Results on Schwarz Alternating Methodsp. 27
Analysis for the Case of Two Subdomainsp. 27
The Case of More than Two Subdomainsp. 29
Abstract Theory of Schwarz Methodsp. 35
Introductionp. 35
Schwarz Methodsp. 35
Convergence Theoryp. 39
Historical Remarksp. 46
Additional Resultsp. 46
Coloring Techniquesp. 46
A Hybrid Methodp. 47
Comparison Resultsp. 51
Remarks on the Implementationp. 52
Two-Level Overlapping Methodsp. 55
Introductionp. 55
Local Solversp. 56
A Coarse Problemp. 59
Scaling and Quotient Space Argumentsp. 60
Technical Toolsp. 62
Convergence Resultsp. 67
Remarks on the Implementationp. 70
Numerical Resultsp. 73
Restricted Schwarz Algorithmsp. 75
Alternative Coarse Problemsp. 75
Convergence Resultsp. 76
Smoothed Aggregation Techniquesp. 81
Partition of Unity Coarse Spacesp. 84
Substructuring Methods: Introductionp. 87
Introductionp. 87
Problem Setting and Geometryp. 88
Schur Complement Systemsp. 94
Discrete Harmonic Extensionsp. 96
Condition Number of the Schur Complementp. 97
Technical Toolsp. 99
Interpolation into Coarse Spacesp. 99
Inequalities for Edgesp. 101
Inequalities for Facesp. 105
Inequalities for Vertices and Auxiliary Resultsp. 111
Primal Iterative Substructuring Methodsp. 113
Introductionp. 113
Local Design and Analysisp. 113
Local Solversp. 115
Coarse Spaces and Condition Number Estimatesp. 117
Vertex Based Methodsp. 118
Wire Basket Based Algorithmsp. 123
Face Based Algorithmsp. 126
Neumann-Neumann and FETI Methodsp. 131
Introductionp. 131
Balancing Neumann-Neumann Methodsp. 133
Definition of the Algorithmp. 133
Matrix Form of the Algorithmp. 137
Condition Number Boundsp. 139
One-Level FETI Methodsp. 143
A Review of the One-Level FETI Methodsp. 144
The Case of Nonredundant Lagrange Multipliersp. 150
The Case of Redundant Lagrange Multipliersp. 156
Dual-Primal FETI Methodsp. 160
FETI-DP Methods in Two Dimensionsp. 161
A Family of FETI-DP Algorithms in Three Dimensionsp. 167
Analysis of Three FETI-DP Algorithmsp. 175
Implementation of FETI-DP Methodsp. 185
Computational Resultsp. 187
Spectral Element Methodsp. 193
Introductionp. 193
Deville-Mund Preconditionersp. 196
Two-Level Overlapping Schwarz Methodsp. 198
Iterative Substructuring Methodsp. 200
Technical Toolsp. 202
Algorithms and Condition Number Boundsp. 206
Remarks on p and hp Approximationsp. 210
More General p Approximationsp. 210
Extensions to hp Approximationsp. 214
Linear Elasticityp. 217
Introductionp. 217
A Two-Level Overlapping Methodp. 219
Iterative Substructuring Methodsp. 220
A Wire Basket Based Methodp. 221
An Extension from the Interfacep. 222
An Extension from the Wire Basketp. 222
A Wire Basket Preconditioner for Linear Elasticityp. 224
Neumann-Neumann and FETI Methodsp. 225
A Neumann-Neumann Algorithm for Linear Elasticityp. 225
One-Level FETI Algorithms for Linear Elasticityp. 227
FETI-DP Algorithms for Linear Elasticityp. 227
Preconditioners for Saddle Point Problemsp. 231
Introductionp. 231
Block Preconditionersp. 235
Flows in Porous Mediap. 239
Iterative Substructuring Methodsp. 241
Hybrid-Mixed Formulations and Spectral Equivalencies with Crouzeix-Raviart Approximationsp. 246
A Balancing Neumann-Neumann Methodp. 250
Overlapping Methodsp. 255
The Stokes Problem and Almost Incompressible Elasticityp. 257
Block Preconditionersp. 258
Iterative Substructuring Methodsp. 261
Computational Resultsp. 269
Problems in H(div ; ¿) and H(curl; ¿)p. 271
Overlapping Methodsp. 274
Problems in H(curl; ¿)p. 276
Problems in H(div; ¿)p. 283
Final Remarks on Overlapping Methods and Numerical Resultsp. 286
Iterative Substructuring Methodsp. 288
Technical Toolsp. 291
A Face-Based Methodp. 299
A Neumann-Neumann Methodp. 301
Remarks on Two-Dimensional Problems and Numerical Resultsp. 305
Iterative Substructuring for Nédélec Approximations in Three Dimensionsp. 308
Indefinite and Nonsymmetric Problemsp. 311
Introductionp. 311
Algorithms on Overlapping Subregionsp. 314
An Iterative Substructuring Methodp. 320
Numerical Resultsp. 321
A Nonsymmetric Problemp. 322
The Helmholtz Equationp. 324
A Variable-Coefficient, Nonsymmetric Indefinite Problemp. 324
Additional Topicsp. 326
Convection-Diffusion Problemsp. 326
The Helmholtz Equationp. 330
Optimized Interface Conditionsp. 333
Nonlinear and Eigenvalue Problemsp. 334
Elliptic Problems and Sobolev Spacesp. 337
Sobolev Spacesp. 337
Trace Spacesp. 341
Linear Operatorsp. 343
Poincaré and Friedrichs Type Inequalitiesp. 343
Spaces of Vector-Valued Functionsp. 346
The Space H(div; ¿)p. 347
The Space H(curl; ¿) in Two Dimensionsp. 348
The Space H(curl; ¿) in Three Dimensionsp. 349
The Kernel and Range of the Curl and Divergence Operatorsp. 350
Positive Definite Problemsp. 353
Scalar Problemsp. 355
Linear Elasticityp. 357
Problems in H(div; ¿) and H(curl; ¿)p. 360
Non-Symmetric and Indefinite Problemsp. 362
Generalizations of the Lax-Milgram Lemmap. 362
Saddle-Point Problemsp. 364
Regularity Resultsp. 369
Galerkin Approximationsp. 371
Finite Element Approximationsp. 371
Triangulationsp. 371
Finite Element Spacesp. 372
Symmetric, Positive Definite Problemsp. 374
Non-Symmetric and Indefinite Problemsp. 375
Spectral Element Approximationsp. 376
Divergence and Curl Conforming Finite Elementsp. 380
Raviart-Thomas Elementsp. 380
Nédélec Elements in Two Dimensionsp. 382
Nédélec Elements in Three Dimensionsp. 383
The Kernel and Range of the Curl and Divergence Operatorsp. 384
Saddle-Point Problemsp. 386
Finite Element Approximations for the Stokes Problemp. 387
Spectral Element Approximations for the Stokes Problemp. 388
Finite Element Approximations for Flows in Porous Mediap. 389
Inverse Inequalitiesp. 389
Matrix Representation and Condition Numberp. 390
Solution of Algebraic Linear Systemsp. 395
Eigenvalues and Condition Numberp. 395
Direct Methodsp. 397
Factorizationsp. 397
Fill-inp. 398
Richardson Methodp. 399
Steepest Descentp. 402
Conjugate Gradient Methodp. 403
Methods for Non-Symmetric and Indefinite Systemsp. 407
The Generalized Minimal Residual Methodp. 407
The Conjugate Residual Methodp. 409
Referencesp. 413
Indexp. 447
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540206965
ISBN-10: 3540206965
Series: Springer Series in Computational Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 450
Published: 1st November 2004
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 0.78