Corrector Theory for MsFEM and HMM in Random Media

Abstract. We analyze the random fluctuations of several multi-scale algorithms such as the multi-scale finite element method (MsFEM) and the finite element heterogeneous multi-scale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multi-scale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multi-scale algorithms in the presence of random environments with short-range and long-range correlations. For a given mesh size h, we show that the fluctuations converge in distribution in the space of continuous paths to Gaussian processes as the correlation length ε → 0. We next derive the limit of such Gaussian processes as h → 0 and compare this limit with the distribution of the random fluctuations of the continuous model. When such limits agree, we conclude that the multi-scale algorithm captures the random fluctuations accurately and passes the corrector test. This property serves as an interesting Benchmark to assess the behavior of the multi-scale algorithm in practical situations where the assumptions necessary for the theory of homogenization are not met. What we find is that the computationally more expensive methods MsFEM, and HMM with a choice of parameter δ = h, correctly capture the random fluctuations both for short-range and longrange oscillations in the medium. The less expensive method HMM with δ < h correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.