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Multiscale Finite Element Methods for Elliptic Problems with Randomly Perturbed Data

Abstract. We propose a method for efficient solution of elliptic problems with multiscale features and randomly perturbed coefficients. We use the multiscale finite element method (MsFEM) as a starting point and derive an algorithm for solving a large number of multiscale problems in parallel. The method is intended to be used within a Monte Carlo framework where solutions corresponding to samples of the randomly perturbed data need to be computed. We show that the proposed method converges to MsFEM solution in the limit for each individual sample of the data. We also show that the complexity of the method is proportional to one solve using MsFEM (where the fine scale is resolved) plus solving N (number of samples) linear systems of equations on the coarse scale, as opposed to solving N problems using MsFEM. A set of numerical examples is presented to illustrate the theoretical findings.

Keywords. Multiscale finite element method, elliptic equation, random perturbation, Neumann series