Domain Decomposition Methods

Abstract

In the previous chapters we have concentrated on the application and theory of spectral methods for problems in simple domains. There have been a number of recent developments on the use of spectral techniques in more general geometries. The basic idea has been to partition the complete domain of the problem into several subdomains. One situation in which this approach is useful is illustrated in Fig. 13.1. The approximation is spectral if increased accuracy is obtained by increasing the order of approximation in a fixed number of subdomains, rather than by resorting to a further partitioning. In this particular example it is clear that at least two subdomains are required in order to use spectral methods at all. Additional advantages may arise by using separate subdomains Ω₂, Ω₃, and Ω₄ instead of a single subdomain which is their union. The partitioning illustrated in Fig. 13.1 leads to a distribution of Chebyshev collocation points which improves the resolution of the approximation. Moreover, one would expect the corresponding algebraic problem to be better-conditioned because there is a less extreme ratio of the largest to smallest grid spacings. Finally, the use of subdomains facilitates the implementation of spectral methods on parallel computers, especially those with local memory.