ANALYSIS OF PRECONDITIONERS FOR DOMAIN DECOMPOSITION.

Domain decomposition is a class of techniques that are designed to solve elliptic problems on irregular domains and on multiprocessor systems. Typically, a domain is decomposed into many smaller regular subdomains and the capacitance system governing the interface unknowns is solved by some version of the preconditioned conjugate gradient method. In this paper, we show that for a simple model problem - Poisson's equation on a rectangle decomposed into two smaller rectangles - the capacitance system can be inverted exactly by Fast Fourier Transform. An exact eigen-decomposition of the capacitance matrix also makes it possible to relate and compare the various preconditioners that have been proposed in the literature.