Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element

Tarek P. Mathew

ISBN 10: 0243091605 / ISBN 13: 9780243091607

Published by Forgotten Books, 2018

About this Item

Print on Demand. This book explores iterative methods to solve saddle point linear systems, focusing on those derived from mixed finite element discretizations of elliptic Neumann problems using Raviart-Thomas elements. The book begins by establishing a theoretical framework. It then provides a proof of convergence of the iterative methods when applied to the mixed finite element case, showing that the rate of convergence is independent of the mesh parameter h. The book goes on to study algorithms involving subdomains with overlap, such as the classical Schwarz alternating method and the additive Schwarz method. It offers proofs of convergence for these iterative methods when applied to the mixed finite element case, again demonstrating that the rate of convergence is independent of h. Finally, the book examines a Dirichlet-Neumann algorithm for the mixed finite element case, providing a proof of convergence showing independence from h. The book concludes by discussing quantitative bounds for some many-level FAC algorithms. The author's insights are significant because they establish the convergence of iterative methods for solving saddle point linear systems arising from mixed finite element discretizations of elliptic Neumann problems, and they show that the rate of convergence is independent of the mesh parameter h. 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 # 9780243091607_0

Bibliographic Details

Title: Domain Decomposition and Iterative ...

Publisher: Forgotten Books

Publication Date: 2018

Binding: Paperback

Condition: New

Book Type: print-on-demand item

About this title

Synopsis:

Excerpt from Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretisations of Elliptic Problems

In this Chapter, we present the relevent background about saddle point formula tions of elliptic problems, and mixed finite element methods using the raviart-thomas spaces. We then discuss an extension theorem for the raviart-thomas finite element spaces, which we use later in establishing bounds for the rate of convergence of the domain decomposition methods. A section containing some background on iterative methods, is also included.

In Chapter 2, we discuss algorithms involving subdomains with overlap, such as the classical Schwarz alternating method, cf. Lions and the additive Schwarz method, as studied by Dryja and Widlund We present proofs of convergenceof these iterative methods when applied to the mixed finite element case, and also show that the rate of convergence is independent of the mesh parmeter h. We also present numerical results of tests using these methods with many subdomains and a certain coarse mesh model problem, which improves the rate of convergence. The results indicate a rate of convergence, which is independent of the mesh parameter h and even the number of subdomains. We have also tested the methods on problems in which the discontinuity in the coefficients of the elliptic operator is large. Such large variations in the coefficients occur in certain applications involving flow in porous media. The rate of convergence remains independent of the jump in the discontinuity, but the accuracy of the pressure deteriorates. See the section on numerical results, in Chapter 2.

About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

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.