Motivation for Preconditioning.- A Finite Element Tutorial.- A Main Goal.- Block Factorization Preconditioners.- Two-by-Two Block Matrices and Their Factorization.- Classical Examples of Block-Factorizations.- Multigrid (MG).- Topics on Algebraic Multigrid (AMG).- Domain Decomposition (DD) Methods.- Preconditioning Nonsymmetric and Indefinite Matrices.- Preconditioning Saddle-Point Matrices.- Variable-Step Iterative Methods.- Preconditioning Nonlinear Problems.- Quadratic Constrained Minimization Problems.
Part I: Motivation for preconditioning. A finite element tutorial. The main goal.- Part II: Block factorization preconditioners. Two-by-two block matrices. Classical examlpes of block factorizations. Multigrid (MG). Topics in algebraic multigrid (AMG). Domain Decomposition (DD) Methods. Preconditioning nonsymmetric and indefinite matrices. Preconditioning saddle-point matrices. Variable-step iterative methods. Preconditioning nonlinear problems. Quadratic constrained minimization problems.- Part III: Appendices. GCG Methods. Properties of finite element matrices. Further details. Computable scales of Sobolev norms. Multilevel algorithms for boundary extension mappings. H01-norm characterization. MG convergence results for finite element problems.- Some auxiliary inequalities.
This monograph is the first to provide a comprehensive, self-contained and rigorous presentation of some of the most powerful preconditioning methods for solving finite element equations in a common block-matrix factorization framework.
The book covers both algorithms and analysis using a common block-matrix factorization approach which emphasizes its unique feature. Topics covered include the classical incomplete block-factorization preconditioners, the most efficient methods such as the multigrid, algebraic multigrid, and domain decomposition.
This text can serve as an indispensable reference for researchers, graduate students, and practitioners. It can also be used as a supplementary text for a topics course in preconditioning and/or multigrid methods at the graduate level.
Uses block-matrix factorization to represent important developments in the field such as the algebraic multi-grid and domain decomposition methods
Rigorous and self-contained presentation
Includes four useful appendices
Excellent reference for practitioners and researchers