Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations

by Mathew, Tarek P. A.

• ISBN13: 9783540772057
• ISBN10: 3540772057
• Format: Paperback
• Copyright: 2008-10-30
• Publisher: Springer Nature

Summary

"Domain decomposition methods are divide and conquer methods for the parallel and computational solution of partial differential equations of elliptic or parabolic type. They include iterative algorithms for solving the discretized equations, techniques for non-matching grid discretizations and techniques for heterogeneous approximations. This book serves as an introduction to this subject, with emphasis on matrix formulations. The topics studied include Schwarz, substructuring, Lagrange multiplier and least squares-control hybrid formulations, multilevel methods, non-self adjoint problems, parabolic equations, saddle point problems (Stokes, porous media and optimal control), non-matching grid discretizations, heterogeneous models, fictitious domain methods, variational inequalities, maximum norm theory, eigenvalue problems, optimization problems and the Helmholtz scattering problem. Selected convergence theory is included."--BOOK JACKET.

Table of Contents

Decomposition Frameworks	p. 1
Hybrid Formulations	p. 2
Schwarz Framework	p. 9
Steklov-Poincare Framework	p. 16
Lagrange Multiplier Framework	p. 27
Least Squares-Control Framework	p. 36
Schwarz Iterative Algorithms	p. 47
Background	p. 48
Projection Formulation of Schwarz Algorithms	p. 56
Matrix Form of Schwarz Subspace Algorithms	p. 66
Implementational Issues	p. 72
Theoretical Results	p. 77
Schur Complement and Iterative Substructuring Algorithms	p. 107
Background	p. 108
Schur Complement System	p. 110
FFT Based Direct Solvers	p. 125
Two Subdomain Preconditioners	p. 140
Preconditioners in Two Dimensions	p. 155
Preconditioners in Three Dimensions	p. 162
Neumann-Neumann and Balancing Preconditioners	p. 175
Implementational Issues	p. 185
Theoretical Results	p. 192
Lagrange Multiplier Based Substructuring: FETI Method	p. 231
Constrained Minimization Formulation	p. 232
Lagrange Multiplier Formulation	p. 239
Projected Gradient Algorithm	p. 241
FETI-DP and BDDC Methods	p. 250
Computational Issues and Parallelization	p. 263
Algorithms for Automated Partitioning of Domains	p. 264
Parallelizability of Domain Decomposition Solvers	p. 280
Least Squares-Control Theory: Iterative Algorithms	p. 295
Two Overlapping Subdomains	p. 296
Two Non-Overlapping Subdomains	p. 303
Extensions to Multiple Subdomains	p. 310
Multilevel and Local Grid Refinement Methods	p. 313
Multilevel Iterative Algorithms	p. 314
Iterative Algorithms for Locally Refined Grids	p. 321
Non-Self Adjoint Elliptic Equations: Iterative Methods	p. 333
Background	p. 334
Diffusion Dominated Case	p. 340
Advection Dominated Case	p. 348
Time Stepping Applications	p. 364
Theoretical Results	p. 366
Parabolic Equations	p. 377
Background	p. 378
Iterative Algorithms	p. 381
Non-Iterative Algorithms	p. 384
Parareal-Multiple Shooting Method	p. 401
Theoretical Results	p. 408
Saddle Point Problems	p. 417
Properties of Saddle Point Systems	p. 418
Algorithms Based on Duality	p. 426
Penalty and Regularization Methods	p. 434
Projection Methods	p. 437
Krylov Space and Block Matrix Methods	p. 445
Applications to the Stokes and Navier-Stokes Equations	p. 456
Applications to Mixed Formulations of Elliptic Equations	p. 474
Applications to Optimal Control Problems	p. 489
Non-Matching Grid Discretizations	p. 515
Multi-Subdomain Hybrid Formulations	p. 516
Mortar Element Discretization: Saddle Point Approach	p. 523
Mortar Element Discretization: Nonconforming Approach	p. 551
Schwarz Discretizations on Overlapping Grids	p. 555
Alternative Nonmatching Grid Discretization Methods	p. 559
Applications to Parabolic Equations	p. 564
Heterogeneous Domain Decomposition Methods	p. 575
Steklov-Poincare Heterogeneous Model	p. 576
Schwarz Heterogeneous Models	p. 585
Least Squares-Control Heterogeneous Models	p. 589
[chi]-Formulation	p. 594
Applications to Parabolic Equations	p. 603
Fictitious Domain and Domain Imbedding Methods	p. 607
Background	p. 608
Preconditioners for Neumann Problems	p. 610
Preconditioners for Dirichlet Problems	p. 611
Lagrange Multiplier and Least Squares-Control Solvers	p. 614
Variational Inequalities and Obstacle Problems	p. 621
Background	p. 622
Projected Gradient and Relaxation Algorithms	p. 628
Schwarz Algorithms for Variational Inequalities	p. 633
Monotone Convergence of Schwarz Algorithms	p. 636
Applications to Parabolic Variational Inequalities	p. 644
Maximum Norm Theory	p. 647
Maximum Principles and Comparison Theorems	p. 648
Well Posedness of the Schwarz Hybrid Formulation	p. 659
Convergence of Schwarz Iterative Algorithms	p. 661
Analysis of Schwarz Nonmatching Grid Discretizations	p. 668
Analysis of Schwarz Heterogeneous Approximations	p. 674
Applications to Parabolic Equations	p. 677
Eigenvalue Problems	p. 679
Background	p. 680
Gradient and Preconditioned Gradient Methods	p. 682
Schur Complement Methods	p. 683
Schwarz Subspace Methods	p. 684
Modal Synthesis Method	p. 686
Optimization Problems	p. 689
Traditional Algorithms	p. 690
Schwarz Minimization Algorithms	p. 697
Helmholtz Scattering Problem	p. 699
Background	p. 700
Non-Overlapping and Overlapping Subdomain Methods	p. 701
Fictitious Domain and Control Formulations	p. 704
Hilbert Uniqueness Method for Standing Waves	p. 705
References	p. 711
Index	p. 761
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