The Non - overlapping Domain

Once a decomposition of the nite element space V into two or more subspaces is given, e.g. , via domain decomposition, a speciic Multiplicative Schwarz Method (MSM) and Additve Schwarz Method (ASM) is deened. In this paper, we analyse the MSM for the decomposition induced by the approximate discrete harmonic nite 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 beneet twice. Indeed, the MSM-DD-preconditioner can be analysed in the MSM framework and implemented as speciic ASM-DD-preconditioner in the parallel PCG method studied previously. Emphazis that we look at the ASM and MSM as techniques for deening 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.