Preconditioned iterative methods for elliptic problems on decomposed domains

A review of preconditioned iterative techniques for elliptic problems on decomposed domains is given. Domain decomposition methods based on overlapping (Schwarz alternating method) and nonoverlapping (Schur complement method) subdomains are described for the model Poisson problem. An additive variant of the Schwarz method which is more suitable to implementation on parallel computers than the traditional multiplicative Schwarz method is presented. Some preconditioners for the Schur domain decomposition method which improve the rate of convergence of the underlying conjugate gradient method are suggested. Some remarks about nonsymmetric problems are made.