Edge-based Schwarz methods for the Crouzeix-Raviart finite volume element discretization of elliptic problems

In this paper, we present two variants of the additive Schwarz method for a Crouzeix-Raviart finite volume element (CRFVE) discretization of second-order elliptic problems with discontinuous coefficients, where the discontinuities may be across subdomain boundaries. The preconditioner in one variant is symmetric, while in the other variant it is nonsymmetric. The proposed methods are quasi optimal, in the sense that the convergence of the preconditioned GMRES iteration in both cases depend only poly-logarithmically on the ratio of the subdomain size to the mesh size. Key words. domain decomposition, Crouzeix-Raviart element, additive Schwarz method, finite volume element, GMRES AMS subject classifications. 65F10, 65N22, 65N30, 63N55