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arXiv · 2109.01616

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

Jianfeng Lu·Felix Otto·Lihan Wang
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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].

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Jianfeng Lu, Felix Otto, Lihan Wang. 2021-09-03. Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media. https://doi.org/10.1080/03605302.2024.2374568

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