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

TL;DR

We address accurate numerical boundary conditions for the gradient of the potential generated by a localized source in a 3D random elliptic medium, with the coefficient field known only inside a box $Q_{2L}$. The main approach uses a multipole expansion grounded in quantitative stochastic homogenization, incorporating first- and second-order correctors (dipole and quadrupole content) via massive approximations and finite-domain truncations, and expressing corrections through a parabolic semigroup framework. The authors show that, under a stationary unit-range ensemble with $\ell\gg 1$ and $L\gg\ell$, the proposed boundary data yields an output $u^{(L)}$ that matches the true solution $u$ with a near-CLT scaling: for $R\in[r_{**},L]$, $(\fint_{B_R} |\nabla(u^{(L)}-u)|^2)^{1/2} \le C (\ell/L)^d (r_{**}/L)^\beta$ with any $\beta<\tfrac{3}{2}$ in 3D (and $\beta$ approaching 2 in higher dimensions). A random radius $r_{**}$ captures the onset of homogenization with controlled tails, while a deterministic, boundary-data based algorithm requires only $a|_{Q_{2L}}$ information. The results extend earlier 2D work to 3D by incorporating quadrupole corrections and establishing rigorous probabilistic and analytic bounds, yielding near-optimal, practically implementable boundary conditions for stochastic media. The work has direct implications for efficient numerical homogenization and boundary-value computations in composite materials and random 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].

Figures (2)

• Figure 1: Numerical convergence rate of $\lvert \nabla u^{(2L)}(\frac{L}{2},\frac{L}{2},\frac{L}{2})-\nabla u^{(L)}(\frac{L}{2},\frac{L}{2},\frac{L}{2}) \rvert$ for the proposed Algorithm \ref{['alg:truealg']}, an algorithm with zero Dirichlet boundary condition, an algorithm without multipole corrections (defined in \ref{['eqn:numnopole']}), and an algorithm with only dipole corrections (defined in \ref{['eqn:numdipole']}). The two figures correspond to two independent realizations of the random media and the same r.h.s.
• Figure 2: $L^2$-averages of $\phi$ and fluctuations of $\psi$. The two figures correspond to independent realizations of the random media.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Corollary 2.2
• Proposition 2.3
• Proposition 2.4
• Remark 2.5
• Remark 2.6
• Lemma 2.7
• Proposition 2.8