Chapter 16 - Some Domain Decomposition Algorithms for Elliptic Problems

Abstract

We discuss domain decomposition methods with which the often very large linear systems of algebraic equations, arising when elliptic problems are discretized by finite differences or finite elements, can be solved with the aid of exact or approximate solvers for the same equations restricted to subregions. The interaction between the subregions, to enforce appropriate continuity requirements, is handled by an iterative method, often a preconditioned conjugate gradient method. Much of the work is local and can be carried out in parallel. We first explore how ideas from structural engineering computations naturally lead to certain matrix splittings. In preparation for the detailed design and analysis of related domain decomposition methods, we then consider the Schwarz alternating algorithm, discovered in 1869. That algorithm can conveniently be expressed in terms of certain projections. We develop these ideas further and discuss an interesting additive variant of the Schwarz method. This also leads to the development of a general framework, which already has proven quite useful in the study of a variety of domain decomposition methods and certain related algorithms. We demonstrate this by developing several algorithms and by showing how their rates of convergence can be estimated. One of them is a Schwarz-type method, for which the subregions overlap, while the others are so called iterative substructuring methods, where the subregions do not overlap. Compared to previous studies of iterative substructuring methods, our proof is simpler and in one case it can be completed without using a finite element extension theorem. Such a theorem has, to our knowledge, always been used in the previous analysis in all but the very simplest cases.