Dual-based {\itshape a posteriori} error estimate for stochastic finite element methods

Abstract

We present an a posteriori error estimation for the numerical solution of a stochastic variational problem arising in the context of parametric uncertainties. The discretization of the stochastic variational problem uses standard finite elements in space and piecewise continuous orthogonal polynomials in the stochastic domain. The a posteriori methodology is derived by measuring the error as the functional difference between the continuous and discrete solutions. This functional difference is approximated using the discrete solution of the primal stochastic problem and two discrete adjoint solutions (on two imbricated spaces) of the associated dual stochastic problem. The dual problem being linear, the error estimation results in a limited computational overhead. With this error estimate, different adaptive refinement strategies of the approximation space can be thought of: applied to the spatial and/or stochastic approximations, by increasing the approximation order or using a finer mesh. In order to investigate the efficiency of different refinement strategies, various tests are performed on the uncertain Burgers’ equation. The lack of appropriate anisotropic error estimator is particularly underlined.