Springer Computational Mathematics

By: Andrea Toselli, Olof Widlund

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Hardcover | 18 October 2004

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468 Pages

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23.5 x 15.88 x 2.54

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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.

Industry Reviews

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)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Probability & Statistics Computing & I.T.Computer Science Computer Architecture & Logic Design Numerical Analysis Applied Mathematics Computer Programming & Software Development Software Engineering

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

Springer Computational Mathematics

At a Glance

Hardcover

Industry Reviews

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