hp-FEM for second moments of elliptic PDEs with stochastic data II: Exponential convergence for stationary singular covariance functions

Abstract

We prove exponential rates of convergence of a class of $hp$ Galerkin Finite Element approximations of solutions to a model tensor nonhypoelliptic equation in the unit square □ = (0,1)$^2$ which exhibit singularities on ∂□ and on the diagonal Δ = {($x,y$) ∈ □ : $x$ = $y$}, but are otherwise analytic in □. As we explained in the first part (Pentenrieder and Schwab, Research Report, Seminar for Applied Mathematics, 2010) of this work, such problems arise as deterministic second moment equations of linear, second order elliptic operator equations $Au = f$ with Gaussian random field data $f$.