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].

翻译：我们关注于计算由尺度为$\ell$的局域电荷分布在无限非均匀介质中产生的电场的数值算法，其中介质仅在围绕电荷支撑的直径为$L\gg\ell$的盒子内已知。我们提出了一种边界条件，在介质为具有有限依赖范围（设定为1且假设$\ell \gg 1$）的平稳系综样本的情境下，该条件以压倒性概率在$\ell$和$L$的标度方面（接近）最优。该边界条件源于允许多极展开的定量随机均匀化理论[BGO20]。本文推广了[LO21]的工作，后者中的算法在二维情形下最优，因此我们除了偶极子外还需考虑四极子。这依次依赖于对一阶校正子之后的二阶校正子的随机估计。基于[GO15]半群方法的推广，我们为所考虑的有限范围系综提供了这些估计。