Domain decomposition and iterative refinement methods for mixed finite element discretisations of elliptic problems.

摘要

n this thesis, we first study the classical Schwarz alternating method and an additive, more parallel variant of it, known as the additive Schwarz method, applied to solve saddle point linear systems obtained by discretising a saddle point formulation of elliptic Neumann problems. We assume that the discretisation is obtained by using a mixed finite element method, in particular the Raviart-Thomas elements. We prove convergence with a rate independent of the mesh parameter h. We also present results of numerical experiments using these algorithms.;Following that, we study two algorithms to solve problems on iteratively refined meshes, namely the fast adaptive composite grid method (FAC), and the asynchronous fast adaptive composite grid method (AFAC). We give a proof of convergence of both these methods in the mixed finite element case, with a rate of convergence independent of the mesh parameters ;Finally, we study a Dirichlet-Neumann type algorithm for the mixed finite element case involving two non-overlapping subdomains. We use this as a preconditioner for a reduced Schur complement system obtained by using an algorithm of Glowinski and Wheeler. We prove that the rate of convergence is independent of