Domain Decomposition and Subspace Methods

Abstract

Domain decomposition is an umbrella term collecting various methods for solving discretised boundary value problems in a domain $\Omega$ by means of a decomposition of $\Omega$. Often this approach is chosen to support parallel computing. After general remarks in Section 12.1, the algorithm using overlapping subdomains is described in Section 12.2. In the case of nonoverlapping subdomains, one needs more involved methods (cf. Section 12.3). In particular the so-called FETI method described in $\S$ 12.3.2 is very popular. The Schur complement method in Section 12.4 gives rise to many variants of iterations. The more abstract view of domain decomposition methods replaces the subdomain by a subspace. Section 12.5 formulates the setting of subspace iterations. Here we distinguish between the additive and multiplicative subspace iteration as explained in the corresponding Sections 12.6 and 12.7. Illustrations follow in Section 12.8. Interestingly, multigrid iterations can also be considered as subspace iterations as analysed in Section 12.9.