Error Control in Multi-Element Generalized Polynomial Chaos Method for Elliptic Problems with Random Coefficients

Abstract

We develop the theory for a robust and efficient adaptive multi-element generalized polynomial chaos (ME-gPC) method for elliptic equations with random coefficients for a moderate number (O(10)) of random dimensions. We employ low-order (p ≤ 3) polynomial chaos and refine the solution using adaptivity in the parametric space. We first study the approximation error of ME-gPC and prove its hp-convergence. We subsequently generate local and global a posteriori error estimators. In order to resolve the error equations efficiently, we construct a reduced space using much smaller number of terms in the enhanced polynomial chaos space to capture the errors of ME-gPC approximation. Based on the a posteriori estimators, we propose and implement an adaptive ME-gPC algorithm for elliptic problems with random coefficients. Numerical results for convergence and efficiency are also presented.