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

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].

References in corpus (13)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• Quantitative stochastic homogenization and large-scale regularity
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• The additive structure of elliptic homogenization
• Quantitative results on the corrector equation in stochastic homogenization
• Correlation structure of the corrector in stochastic homogenization
• Mesoscopic higher regularity and subadditivity in elliptic homogenization
• Quantitative estimates in stochastic homogenization for correlated coefficient fields
• Higher-order pathwise theory of fluctuations in stochastic homogenization
• The choice of representative volumes in the approximation of effective properties of random materials
• Some variance reduction methods for numerical stochastic homogenization
• On the averaged Green's function of an elliptic equation with random coefficients
• On a homogenization problem

Cited by in corpus (1)

• Random acoustic boundary conditions and Weyl's law for Laplace-Beltrami operators on non-smooth boundaries