Mathew Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations

1. Auflage 2008
ISBN: 978-3-540-77209-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 61, 770 Seiten

Reihe: Lecture Notes in Computational Science and Engineering

Domain decomposition methods are divide and conquer computational methods for the parallel solution of partial differential equations of elliptic or parabolic type. The methodology includes iterative algorithms, and techniques for non-matching grid discretizations and heterogeneous approximations. This book serves as a matrix oriented introduction to domain decomposition methodology. A wide range of topics are discussed include hybrid formulations, Schwarz, and many more.

Autoren/Hrsg.

Mathew, Tarek

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;1 Decomposition Frameworks;13
3.1;1.1 Hybrid Formulations;14
3.2;1.2 Schwarz Framework;21
3.3;1.3 Steklov-Poincar´e Framework;28
3.4;1.4 Lagrange Multiplier Framework;39
3.5;1.5 Least Squares-Control Framework;48
4;2 Schwarz Iterative Algorithms;59
4.1;2.1 Background;60
4.2;2.2 Projection Formulation of Schwarz Algorithms;68
4.3;2.3 Matrix Form of Schwarz Subspace Algorithms;78
4.4;2.4 Implementational Issues;84
4.5;2.5 Theoretical Results;89
5;3 Schur Complement and Iterative Substructuring Algorithms;118
5.1;3.1 Background;119
5.2;3.2 Schur Complement System;121
5.3;3.3 FFT Based Direct Solvers;136
5.4;3.4 Two Subdomain Preconditioners;151
5.5;3.5 Preconditioners in Two Dimensions;166
5.6;3.6 Preconditioners in Three Dimensions;173
5.7;3.7 Neumann-Neumann and Balancing Preconditioners;186
5.8;3.8 Implementational Issues;196
5.9;3.9 Theoretical Results;203
6;4 Lagrange Multiplier Based Substructuring: FETI Method;242
6.1;4.1 Constrained Minimization Formulation;243
6.2;4.2 Lagrange Multiplier Formulation;250
6.3;4.3 Projected Gradient Algorithm;252
6.4;4.4 FETI-DP and BDDC Methods;261
7;5 Computational Issues and Parallelization;274
7.1;5.1 Algorithms for Automated Partitioning of Domains;275
7.2;5.2 Parallelizability of Domain Decomposition Solvers;291
8;6 Least Squares-Control Theory: Iterative Algorithms;306
8.1;6.1 Two Overlapping Subdomains;307
8.2;6.2 Two Non-Overlapping Subdomains;314
8.3;6.3 Extensions to Multiple Subdomains;321
9;7 Multilevel and Local Grid Re.nement Methods;324
9.1;7.1 Multilevel Iterative Algorithms;325
9.2;7.2 Iterative Algorithms for Locally Re.ned Grids;332
10;8 Non-Self Adjoint Elliptic Equations: Iterative Methods;343
10.1;8.1 Background;344
10.2;8.2 Di.usion Dominated Case;350
10.3;8.3 Advection Dominated Case;358
10.4;8.4 Time Stepping Applications;374
10.5;8.5 Theoretical Results;376
11;9 Parabolic Equations;387
11.1;9.1 Background;388
11.2;9.2 Iterative Algorithms;391
11.3;9.3 Non-Iterative Algorithms;394
11.4;9.4 Parareal-Multiple Shooting Method;411
11.5;9.5 Theoretical Results;418
12;10 Saddle Point Problems;427
12.1;10.1 Properties of Saddle Point Systems;428
12.2;10.2 Algorithms Based on Duality;436
12.3;10.3 Penalty and Regularization Methods;444
12.4;10.4 Projection Methods;447
12.5;10.5 Krylov Space and Block Matrix Methods;455
12.6;10.6 Applications to the Stokes and Navier-Stokes Equations;466
12.7;10.7 Applications to Mixed Formulations of Elliptic Equations;484
12.8;10.8 Applications to Optimal Control Problems;499
13;11 Non-Matching Grid Discretizations;524
13.1;11.1 Multi-Subdomain Hybrid Formulations;525
13.2;11.2 Mortar Element Discretization: Saddle Point Approach;532
13.3;11.3 Mortar Element Discretization: Nonconforming Approach;560
13.4;11.4 Schwarz Discretizations on Overlapping Grids;564
13.5;11.5 Alternative Nonmatching Grid Discretization Methods;568
13.6;11.6 Applications to Parabolic Equations;573
14;12 Heterogeneous Domain Decomposition Methods;583
14.1;12.1 Steklov-Poincar´e Heterogeneous Model;584
14.2;12.2 Schwarz Heterogeneous Models;593
14.3;12.3 Least Squares-Control Heterogeneous Models;597
14.4;12.4 .-Formulation;602
14.5;12.5 Applications to Parabolic Equations;611
15;13 Fictitious Domain and Domain Imbedding Methods;615
15.1;13.1 Background;616
15.2;13.2 Preconditioners for Neumann Problems;618
15.3;13.3 Preconditioners for Dirichlet Problems;619
15.4;13.4 Lagrange Multiplier and Least Squares-Control Solvers;622
16;14 Variational Inequalities and Obstacle Problems;628
16.1;14.1 Background;629
16.2;14.2 Projected Gradient and Relaxation Algorithms;635
16.3;14.3 Schwarz Algorithms for Variational Inequalities;640
16.4;14.4 Monotone Convergence of Schwarz Algorithms;643
16.5;14.5 Applications to Parabolic Variational Inequalities;651
17;15 Maximum Norm Theory;654
17.1;15.1 Maximum Principles and Comparison Theorems;655
17.2;15.2 Well Posedness of the Schwarz Hybrid Formulation;666
17.3;15.3 Convergence of Schwarz Iterative Algorithms;668
17.4;15.4 Analysis of Schwarz Nonmatching Grid Discretizations;675
17.5;15.5 Analysis of Schwarz Heterogeneous Approximations;681
17.6;15.6 Applications to Parabolic Equations;684
18;16 Eigenvalue Problems;686
18.1;16.1 Background;687
18.2;16.2 Gradient and Preconditioned Gradient Methods;689
18.3;16.3 Schur Complement Methods;690
18.4;16.4 Schwarz Subspace Methods;691
18.5;16.5 Modal Synthesis Method;693
19;17 Optimization Problems;695
19.1;17.1 Traditional Algorithms;696
19.2;17.2 Schwarz Minimization Algorithms;703
20;18 Helmholtz Scattering Problem;705
20.1;18.1 Background;706
20.2;18.2 Non-Overlapping and Overlapping Subdomain Methods;707
20.3;18.3 Fictitious Domain and Control Formulations;710
20.4;18.4 Hilbert Uniqueness Method for Standing Waves;711
21;References;716
22;Index;766