Sparse high order FEM for elliptic sPDEs

Abstract

We describe the analysis and the implementation of two finite element (FE) algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients. They are based on separation of deterministic and stochastic parts of the input data by a Karhunen–Loève expansion, truncated after M terms. With a change of measure we convert the problem to a sequence of M-dimensional, parametric deterministic problems. Two sparse, high order polynomial approximations of the random solution’s joint pdf’s, parametrized in the input data’s Karhunen–Loève expansion coordinates, are analyzed: a sparse stochastic Galerkin FEM (sparse sGFEM) and a sparse stochastic Collocation FEM (sparse sCFEM). A priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution’s joint pdf’s to the stochastic regularity of the input data. sCFEM and sGFEM yield deterministic approximations of the random solutions joint pdf’s that converge spectrally in the number of deterministic problems to be solved. Numerical examples with random inputs of small correlation length in diffusion problems are presented. High order gPC approximations of solutions with stochastic parameter spaces of dimension up to M = 80 are computed on workstations.