Sparse finite elements for stochastic elliptic problems - higher order moments

Abstract

We define the higher order moments associated to the stochastic solution of an elliptic BVP in D \subset Rd with stochastic source terms and boundary data. We prove that the k-th moment (or k-point correlation function) of the random solution solves a deterministic problem in Dk \subset Rdk. We discuss well-posedness and regularity in scales of Sobolev spaces with bounded mixed derivatives. We discretize this deterministic k-th moment problem using sparse tensor product FE-spaces and, exploiting a spline wavelet basis, we propose an algorithm of (up to logarithmic terms) the same accuracy and complexity as a multigrid finite element method for the mean field problem in D.