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

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We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L ≫ l around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of l and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that l ≫ 1 ). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion. This work extends, the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of Gloria and Otto.