The optimised Schwarz method and the two-Lagrange multiplier method for heterogeneous problems

Abstract

In modern science and engineering there exist many heterogeneous problems, in which the material under consideration has non-uniform properties. For example when considering seepage under a dam, water will flow at vastly different rates through sand and stone. Mathematically this can be represented as an elliptic boundary value problem that has a large jump in coefficients between subdomains. The optimised Schwarz method and the related two-Lagrange multiplier method are non-overlapping domain decomposition methods that can be used to numerically solve such boundary value problems. These methods work by solving local Robin problems on each subdomain in parallel, which then piece together to give an approximate solution to the global boundary value problem. It is known that with a careful choice of Robin parameter the convergence of these methods can be sped up. In this thesis we first review the known results for the optimised Schwarz method, deriving optimised Robin parameters and studying the asymptotic performance of the method as the mesh parameter of the discretisation is refined and the jump in coefficients becomes large. Next we formulate the two-Lagrange multiplier method for a model two subdomain problem and show its equivalence to the optimised Schwarz method under suitable conditions. The two-Lagrange multiplier method results in a non-symmetric linear system which is usually solved with a Krylov subspace method such as GMRES. The convergence of the GMRES method can be estimated by constructing a conformal map from the exterior of the field of values of the system matrix to the interior of the unit disc. We approximate the field of values of the two-Lagrange multiplier system matrix by a rectangle and calculate optimised Robin parameters that ensure the rectangle 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 is refined and the jump in coefficients becomes large. The final part of the thesis is concerned with the case of heterogeneous problems with many subdomains and cross points, where three or more subdomains coincide. We formulate the two-Lagrange multiplier method for such problems and consider known preconditioners that are needed to improve convergence as the number of subdomains increases. Throughout the thesis numerical experiments are performed to verify the theoretical results.