Sparse Finite Elements for Elliptic Problems with Stochastic Data

Abstract

We formulate elliptic boundary value problems with stochastic loading in a domain D. We show well-posedness of the problem in stochastic Sobolev spaces and we derive then a deterministic elliptic PDE in DxD for the spatial correlation of the solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic FE space in D yields optimal asymptotic rates of convergence for the second moments even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in DxD allows iterative solution of the discrete equations for the correlation kernel in essentially the same complexity as the solution of the mean field equation.