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This paper analyzes the random fluctuations obtained by a heterogeneousmulti-scale first-order finite element method applied to solve ellipticequations with a random potential. We show that the random fluctuations of suchsolutions are correctly estimated by the heterogeneous multi-scale algorithmwhen appropriate fine-scale problems are solved on subsets that cover the wholecomputational domain. However, when the fine-scale problems are solved overpatches that do not cover the entire domain, the random fluctuations may or maynot be estimated accurately. In the case of random potentials with short-rangeinteractions, the variance of the random fluctuations is amplified as theinverse of the fraction of the medium covered by the patches. In the case ofrandom potentials with long-range interactions, however, such an amplificationdoes not occur and random fluctuations are correctly captured independent ofthe (macroscopic) size of the patches.  These results are consistent with those obtained by the authors for moregeneral equations in the one-dimensional setting and provide indications on theloss in accuracy that results from using coarser, and hence lesscomputationally intensive, algorithms.

Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential