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We are interested in numerical algorithms for computing the electrical fieldgenerated by a charge distribution localized on scale $\ell$ in an infiniteheterogeneous medium, in a situation where the medium is only known in a box ofdiameter $L\gg\ell$ around the support of the charge. We propose a boundarycondition that with overwhelming probability is (near) optimal with respect toscaling in terms of $\ell$ and $L$, in the setting where the medium is a samplefrom a stationary ensemble with a finite range of dependence (set to be unityand with the assumption that $\ell \gg 1$). The boundary condition is motivatedby quantitative stochastic homogenization that allows for a multipole expansion[BGO20].  This work extends [LO21], the algorithm in which is optimal in two dimension,and thus we need to take quadrupoles, next to dipoles, into account. This inturn relies on stochastic estimates of second-order, next to first-order,correctors. These estimates are provided for finite range ensembles underconsideration, based on an extension of the semi-group approach of [GO15].

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media