Restricted Additive Schwarz Method for Some Inequalities Perturbed by a Lipschitz Operator

Abstract

We introduce and analyze a restricted additive Schwarz method for some inequalities perturbed by a Lipschitz operator. An existence and uniqueness result concerning the solution of the inequalities we consider is given. Also, we introduce the method as a subspace correction algorithm, prove the convergence and estimate the error in a general framework of a finite dimensional Hilbert space. By introducing the finite element spaces, we get that our algorithm is really a restricted additive Schwarz method and conclude that the convergence condition and convergence rate are independent of the mesh parameters and of both the number of subdomains and the parameters of the domain decomposition, but the convergence condition is a little more restrictive than the existence and uniqueness condition of the solution.