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Domain decomposition algorithms

Published online by Cambridge University Press:  07 November 2008

Tony F. Chan  and
Tarek P. Mathew
Tony F. Chan
Affiliation:
Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90024, USA Email: chan@math.ucla.edu.
Tarek P. Mathew
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036, USA Email: mathew@corral.uwyo.edu.
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Abstract

Domain decomposition refers to divide and conquer techniques for solving partial differential equations by iteratively solving subproblems defined on smaller subdomains. The principal advantages include enhancement of parallelism and localized treatment of complex and irregular geometries, singularities and anomalous regions. Additionally, domain decomposition can sometimes reduce the computational complexity of the underlying solution method.

In this article, we survey iterative domain decomposition techniques that have been developed in recent years for solving several kinds of partial differential equations, including elliptic, parabolic, and differential systems such as the Stokes problem and mixed formulations of elliptic problems. We focus on describing the salient features of the algorithms and describe them using easy to understand matrix notation. In the case of elliptic problems, we also provide an introduction to the convergence theory, which requires some knowledge of finite element spaces and elementary functional analysis.

Type
Research Article
Information
Acta Numerica , Volume 3 , January 1994 , pp. 61 - 143
Copyright
Copyright © Cambridge University Press 1994

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