Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations

TLDR

The study develops two numerical methods to approximate statistical moments, especially the expected value, of a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. The authors employ two approaches: a Monte Carlo scheme that samples stochastic coefficients and solves the resulting deterministic elliptic equations with a standard Galerkin finite element method to compute sample averages, and a deterministic parametric method that approximates the stochastic coefficients in a finite-dimensional space and then applies a Galerkin finite element (h‑ or p‑version) to obtain the desired statistics. The error analysis and computational cost comparison indicate conditions that guide the optimal choice between the two numerical approximations.

Abstract

We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.