Finite elements for elliptic problems with stochastic coefficients

Abstract

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(NlogN) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M,1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r=(r1,…,rM) in an (M,r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r1 up to 10 in moderate time.