Iterative solvers for the stochastic finite element method

Abstract

The stochastic finite element method is a recent technique for solving partial differential equations (PDE) that contain stochastic coefficients. It can result in high-order accurate stochastic solutions while avoiding the computation of numerous Monte Carlo simulations. Two prominent variants are the stochastic collocation and stochastic Galerkin technique. The former samples a stochastic PDE in a set of well-chosen collocation points and results in a number of decoupled deterministic PDEs. The latter transforms a stochastic PDE into a coupled system of deterministic PDEs. We shall illustrate the action of both techniques in the context of a nonlinear PDE with stochastic coefficients. The applicability of the stochastic Galerkin method strongly depends on efficient solvers for the corresponding high-dimensional algebraic systems. In this talk, we focus on a comparison and analysis of iterative solution methods for stochastic Galerkin discretizations of linear stochastic PDEs [1]. The tensor product structure of the algebraic systems is exploited in both one-level as well as multilevel iterative methods. Using multigrid techniques, a convergence rate independent of the stochastic and spatial discretization parameters can be obtained. This may lead however for certain types of problems to a larger solution time than, for example, a Krylov solver with mean-based preconditioner. Numerical experiments illustrate the performance of the iterative solvers for some non-trivial problems. [1] E. Rosseel and S. Vandewalle. Iterative solvers for the stochastic finite element method. SIAM J. Sci. Comput., 2009. Submittedstatus: publishe