On the convergence of Algebraic Optimizable Schwarz Methods with applications to elliptic problems

The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Schwarz-Robin methods use Robin conditions on the artificial interfaces for information exchange at each iteration. Optimized Schwarz Methods (OSM) are those in which one optimizes the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Schwarz-Robin methods still lack a complete theory. In this paper, an abstract Hilbert space version of the OSM is presented, together with an analysis of conditions for its convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Schwarz-Robin methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence rate ω(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence rate does not appear to depend on h.