A priori error estimation for the stochastic perturbation method

The perturbation method has been among the most popular stochastic finite element methods due to its simplicity and efficiency. The error estimation for the perturbation method is well established for deterministic problems, but until now there has not been an error estimation developed in the probabilistic context. This paper presents a priori error estimation for the perturbation method in solving stochastic partial differential equations. The physical problems investigated here come from linear elasticity of heterogeneous materials, where the material parameters are represented by stochastic fields. After applying the finite element discretization to the physical problem, a stochastic linear algebraic equation system is formed with a random matrix on the left hand side. Such systems have been efficiently solved by using the stochastic perturbation approach, without knowing how accurate/inaccurate the perturbation solution is. In this paper, we propose a priori error estimation to directly link the error of the solution vector with the variation of the source stochastic field. A group of examples are presented to demonstrate the effectiveness of the proposed error estimation.