A finite element method for elliptic problems with stochastic input data

Abstract

We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic loadings. In case of elliptic problems on stochastic domains or with stochastic coefficients analogous expressions hold to leading order in the size of the stochastic perturbation. The solution's two-point correlation satisfies a deterministic tensor product partial differential equation on the twofold product domain. For its numerical solution we apply a sparse tensor product approximation by multilevel frames. This way standard finite element techniques can be used. Numerical examples illustrate feasibility and scope of the method.