MATHICSE Technical Report: A posteriori error estimation for the stochastic collocation finite element approximation of the heat equation with random coefficients

In this work we present a residual based a posteriori error estimation for a heat equation with a random forcing term and a random diffusion coefficient which is assumed to depend affinely on a finite number of independent random variables. The problem is discretized by a stochastic collocation finite element method and advanced in time by the θ-scheme. The a posteriori error estimate consists of three parts controlling the finite element error, the time discretization error and the stochastic collocation error, respectively. These estimators are then used to drive an adaptive choice of FE mesh, collocation points and time steps. We study the effectiveness of the estimate and the performance of the adaptive algorithm on a numerical example.