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. Several multi-scale numerical algorithms have been shown to correctly capture
the homogenized limit of solutions of elliptic equations with coefficients modeled as
stationary and ergodic random fields. Because theoretical results are available in the
continuum setting for such equations, we consider here the case of a second-order elliptic
equations with random potential in two dimensions of space. 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 in [9] for more
general equations in the one-dimensional setting and provide indications on the loss in
accuracy that results from using coarser, and hence computationally less intensive,
algorithms.