On the convergence of Optimized Schwarz Methods by way of Matrix Analysis

Domain decomposition methods are widely used to solve in parallel large linear systems of equations arising in the discretization of partial differential equations. Optimized Schwarz Methods (OSM) have been the subject of intense research because they lead to algorithms that converge very quickly. The analysis of OSM has been a very challenging research area and there are currently no general proofs of convergence for the optimized choices of the Robin parameter in the case of overlap. In this article, we apply a proof technique developed for the analysis of Schwarz-type algorithms using matrix analysis techniques and specifically using properties of matrix splittings, to the Optimized Schwarz algorithms. We thus obtain new general convergence results, but they apply only to large Robin parameters, which may not be the optimal ones.