A multi-modes Monte Carlo finite element method for elliptic partial differential equations with random coefficients

摘要

This paper develops and analyzes an efficient numerical method for solvingelliptic partial differential equations, where the diffusion coefficients arerandom perturbations of deterministic diffusion coefficients. The method isbased upon a multi-modes representation of the solution as a power series ofthe perturbation parameter, and the Monte Carlo technique for sampling theprobability space. One key feature of the proposed method is that the governingequations for all the expanded mode functions share the same deterministicdiffusion coefficients, thus an efficient direct solver by repeated use of the$LU$ decomposition matrices can be employed for solving the finite elementdiscretized linear systems. It is shown that the computational complexity ofthe whole algorithm is comparable to that of solving a few deterministicelliptic partial differential equations using the $LU$ director solver. Errorestimates are derived for the method, and numerical experiments are provided totest the efficiency of the algorithm and validate the theoretical results.