The non-overlapping domain decomposition multiplicative schwarz method

Abstract

Once a decomposition of the finite element space V into two or more subspaces is given, e.g., via domain decomposition, a specific Multiplicative Schwarz Method (MSM) and Additive Schwarz Method (ASM) is defined. In this paper, we analyse the MSM for the decomposition induced by the approximate discrete harmonic finite element basis which was introduced in a joint paper of the authors with A. Meyer (1990). The main theorem of the present paper states that a special symmetric version of the MSM with approximate orthoprojections is equivalent to some ASM with specially chosen basic transformation and block preconditioners. From this observation we can benefit twice. Indeed, the MSM-DD-preconditioner can be analysed in the MSM framework and implemented as specific ASM-DD-preconditioner in the parallel PCG method studied previously. Emphasis that we look at the ASM and MSM as techniques for defining and analysing parallel DD preconditioners used then in a parallelized version of the PCG-method which is well suited for computations on MIMD computers with local memory and message passing principle.