  This paper analyzes the random fluctuations obtained by a heterogeneous
multi-scale first-order finite element method applied to solve elliptic
equations with a random potential. We show that the random fluctuations of such
solutions are correctly estimated by the heterogeneous multi-scale algorithm
when appropriate fine-scale problems are solved on subsets that cover the whole
computational domain. However, when the fine-scale problems are solved over
patches that do not cover the entire domain, the random fluctuations may or may
not be estimated accurately. In the case of random potentials with short-range
interactions, the variance of the random fluctuations is amplified as the
inverse of the fraction of the medium covered by the patches. In the case of
random potentials with long-range interactions, however, such an amplification
does not occur and random fluctuations are correctly captured independent of
the (macroscopic) size of the patches.
  These results are consistent with those obtained by the authors for more
general equations in the one-dimensional setting and provide indications on the
loss in accuracy that results from using coarser, and hence less
computationally intensive, algorithms.

Numerical Analysis;Analysis of PDEs;Probability