On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring

Abstract

In this paper, we consider that the subdomains of the domain decomposition are colored such that the subdomains with the same color do not intersect and introduce and analyze the convergence of a damped additive Schwarz method related to such a subdomain coloring for the resolution of variational inequalities and equations. In this damped method, a single damping value is associated with all the subdomains having the same color. We first make this analysis both for variational inequalities and, as a special case, for equations in an abstract framework. By introducing an assumption on the decomposition of the convex set of the variational inequality, we theoretically analyze in a reflexive Banach space the convergence of the damped additive Schwarz method. The introduced assumption contains a constant C0, and we explicitly write the expression of the convergence rates, depending on the number of colors and the constant C0, and find the values of the damping constants which minimize them. For problems in the finite element spaces, we write the constant C0 as a function of the overlap parameter of the domain decomposition and the number of colors of the subdomains. We show that, for a fixed overlap parameter, the convergence rate, as a function of the number of subdomains has an upper limit which depends only on the number of the colors of the subdomains. Obviously, this limit is independent of the total number of subdomains. Numerical results are in agreement with the theoretical ones. They have been performed for an elasto-plastic problem to verify the theoretical predictions concerning the choice of the damping parameter, the dependence of the convergence on the overlap parameter and on the number of subdomains.