A Posteriori Error Estimation for the Stochastic Collocation Finite Element Method

In this work, we consider an elliptic partial differential equation (PDE) with a random coefficient solved with the stochastic collocation finite element method (SC-FEM). The random diffusion coefficient is assumed to depend in an affine way on independent random variables. We derive a residual-based a posteriori error estimate that is constituted of two parts controlling the SC error and the FE error, respectively. The SC error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are given to illustrate the efficiency of the error estimator and the performance of the adaptive algorithm.