  We are interested in numerical algorithms for computing the electrical field
generated by a charge distribution localized on scale $l$ in an infinite
heterogeneous correlated random medium, in a situation where the medium is only
known in a box of diameter $L\gg l$ around the support of the charge. We show
that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary
conditions motivated by the multipole expansion of Bella, Giunti and Otto,
still performs well in correlated media. With overwhelming probability, we
obtain a convergence rate in terms of $l$, $L$ and the size of the correlations
for which optimality is supported with numerical simulations. These estimates
are provided for ensembles which satisfy a multi-scale logarithmic Sobolev
inequality, where our main tool is an extension of the semi-group estimates
established by the first author. As part of our strategy, we construct
sub-linear second-order correctors in this correlated setting which is of
independent interest.
