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] 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].

翻译：我们感兴趣的是计算电场的数字算法,这种电场是按比例分配,以无限的多元介质计算,在这种情况下,介质只在直径一箱的美元=gg\ell$中知道,在支持充电的周围。我们提议了一个边界条件,即极有可能(接近)以美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=千分率=千分率=千分率=1美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=千分率=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=美元=千分率=美元=美元=在这种情况下,介质的驱动因素是量化的同质化,使多极扩大[LO21]的范围从两个维扩大到三个维度,因此我们需要考虑在迪奥尔特尔兹,然后考虑。这又取决于次次次次次次次次次次次测测测测测测测测次次次次次次次次次次次次测测测测测测测测测测测测的第二次次次次的第二等。这些测测测测测测测算。这些测测测测测测测测测测测测算。