Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations

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.

From the reviews:
“This book by Panayot Vassilevski is the first comprehensive text in the literature on multilevel preconditioners in the formulation of approximate block factorizations. … The presentation is comprehensive and detailed, and the theory is illustrated by classical examples of block factorizations like the block-ILU factorization. … valuable addition to the collection of research books for any researcher working in the field of multilevel methods who is interested in getting a different view on methods he or she knows and has helped to develop and shape.” (Martin J. Gander, Mathematical Reviews, Issue 2010 b)
“This well-written and timely book fills a great need in the literature. It provides a thorough and yet readable treatment of cotemporary theory … . book will be readily accessible to graduate students past introductory analysis and linear algebra. … The book will be a important reference for anyone interested in practical or theoretical aspects of iterative solvers for finite elements, from students to established researchers. Early chapters could serve well as a textbook on the topic.” (Jan Mandel, Zentralblatt MATH, Vol. 1170, 2009)
“The presentation sets out with a tutorial on finite elements, where already properties of matrices which turn out essential later on are exhibited. … this presentation will be of major interest to researchers and advanced students in that field.” (H. Muthsam, Monatshefte für Mathematik, Vol. 157 (1), May, 2009)

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.
Topics covered include the classical incomplete block-factorization preconditioners and the most efficient methods such as the multigrid, algebraic multigrid, and domain decomposition. Additionally, the author discusses preconditioning of saddle-point, nonsymmetric and indefinite problems, as well as preconditioning of certain nonlinear and quadratic constrained minimization problems that typically arise in contact mechanics. The book presents analytical as well as algorithmic aspects.
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 Includes supplementary material: sn.pub/extras