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

TL;DR

This paper develops near-optimal artificial boundary conditions for simulating electrical fields in 3D random media, leveraging second-order correctors and stochastic homogenization to improve accuracy over previous methods.

Contribution

It introduces a new boundary condition based on second-order correctors for 3D random elliptic media, extending previous 2D algorithms to account for quadrupoles.

Findings

Boundary condition is near-optimal with high probability.

Extension of stochastic homogenization estimates to second-order correctors.

Algorithm improves accuracy in 3D heterogeneous media simulations.

Abstract

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $L\gg\ell$ 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 $\ell$ 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 $\ell \gg 1$). The boundary condition is motivated by 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 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].