Iterative Methods for the Stochastic Diffusion Problem

Abstract

It is the purpose of this thesis to develop iterative methods for solving the linear systems that arise from application of the stochastic finite element method to steady-state stochastic diffusion problems. Although the theory herein is sufficiently general to be applicable to a variety of choices for the stochastic finite elements, attention is given to the method of polynomial chaos. For the second-order problem a multigrid algorithm is defined wherein the spatial discretization parameter is varied from grid to grid while the stochastic discretization parameter is held constant. It is demonstrated that the convergence rate of this method is independent of the discretization parameters. For the first-order problem, which produces a linear system that is symmetric and indefinite, the MINRES algorithm is applied with a preconditioner that incorporates a multigrid algorithm. This multigrid algorithm, as for the one applied to the second-order problem, varies the spatial discretization from grid to grid while holding the stochastic discretization parameter constant. Again, it is demonstrated that the convergence rate of this method is independent of the discretization parameters.