The optimised Schwarz method and the two-Lagrange multiplier method for heterogeneous problems in general domains with two general subdomains

Abstract

The optimised Schwarz method and the related two-Lagrange multiplier (2LM) method are nonoverlapping domain decomposition methods which can be used to numerically solve boundary value problems. Local Robin problems are solved on each subdomain in parallel to approximate the global solution, where a careful choice of Robin parameters leads to faster convergence. The 2LM method involves the solution of a nonsymmetric linear system that is usually solved with a Krylov subspace method such as GMRES. The speed of convergence of GMRES can be estimated using a conformal map from the exterior of the field of values of the system matrix to the interior of the unit disc. In this article we consider an elliptic PDE problem with a jump in diffusion coefficients across the interface between the subdomains. We approximate the field of values of the 2LM system matrix by a rectangle, R, in ? and provide optimised Robin parameters that ensure R is "well conditioned" in the sense that GMRES converges quickly. We derive convergence estimates for GMRES and consider the behaviour asymptotically as the mesh size h becomes small and the jump in coefficients becomes large. We observe, for our choice of Robin parameters, that increasing the jump in coefficients increases the convergence rate of GMRES. Numerical experiments are performed to verify the theoretical results.