Some Domain Decomposition Algorithms for Nonselfadjoint Elliptic and Parabolic

Xiao-Chuan Cai

ISBN 10: 1333219601 / ISBN 13: 9781333219604

Published by Forgotten Books, 2018

About this Item

Print on Demand. This book presents numerical methods for solving large linear systems that emerge from finite element discretizations of partial differential equations. The author presents a unified abstract theory for additive Schwarz-type methods for general linear, non-selfadjoint, second order elliptic and parabolic problems. The convergence rate of the algorithms is estimated and shown to be optimal in both the elliptic and parabolic problems in R2 and R3. Numerical experiments are also presented. The algorithms introduced in this book are promising for parallel computation because they involve solving a number of smaller linear systems, which correspond to the restriction of the original problem to subregions, instead of the large original system. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. Seller Inventory # 9781333219604_0

Bibliographic Details

Title: Some Domain Decomposition Algorithms for ...

Publisher: Forgotten Books

Publication Date: 2018

Binding: Paperback

Condition: New

Book Type: print-on-demand item

About this title

Synopsis:

Excerpt from Some Domain Decomposition Algorithms for Nonselfadjoint Elliptic and Parabolic Partial Differential Equations: Technical Report 461; September, 1989

The iterative methods most commonly used are the conjugate gradient method for the symmetric, positive definite case and the generalized conju gate residual methods (gmres) for the general, nonsymmetric case. If the symmetric part of the operator is positive definite, with respect to a suitable inner product, convergence can be guaranteed. In this thesis, the rate of convergence of all algorithms will be estimated. We show that the additive Schwarz algorithm is optimal for both elliptic and parabolic problems in R2 and R3 in the sense that the rate of convergence is independent of both the coarse mesh size, defined by the substructures, and the fine mesh size. The iterative substructuring algorithm is not optimal in the above sense, however, in the R2 case the corresponding rate of convergence depends only mildly on the mesh parameters. A modified additive Schwarz algorithm is also introduced for parabolic problems in R2. The rate of convergence is independent of the fine mesh size.

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This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.