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 [BGO20]. This work extends [LO21] from two to three dimensions, 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 [GO15].
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