hp-FEM for second moments of elliptic PDEs with stochastic data Part 2: Exponential convergence

Abstract

We prove exponential rates of convergence of a class of $hp$ Galerkin Finite Element approximations of solutions to a model tensor non-hypoelliptic equation in the unit square □ = (0,1)$^2$ which exhibit singularities on ∂□ and on the diagonal ∆ = {($x,y$) ∈ □ : $x$ = $y$}, but are otherwise analytic in □. As we explained in the first part [6] 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 $f$.