Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation

Abstract

This work studies a linear elliptic problem with uncertainty. The introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations. The major emphasis of this work is the probabilistic treatment of uncertainty, addressing the problem of solving linear elliptic boundary value problems with stochastic coefficients. If the stochastic coefficients are known functions of a random vector, then the stochastic elliptic boundary value problem is turned into a parametric deterministic one with solution u(y,x), y∈Γ, x∈D, where D⊂Rd, d=1,2,3, and Γ is a high-dimensional cube. In addition, the function u is specified as the solution of a deterministic variational problem over Γ×D. A tensor product finite element method, of h-version in D and k-, or, p-version in Γ, is proposed for the approximation of u. A priori error estimates are given and an adaptive algorithm is also proposed. Due to the high dimension of Γ, the Monte Carlo finite element method is also studied here. This work compares the asymptotic complexity of the numerical methods, and shows results from numerical experiments. Comments on the uncertainty in the probabilistic characterization of the coefficients in the stochastic formulation are included.