Artificial boundary conditions for random ellitpic systems with correlated coefficient field

TL;DR

The paper develops and analyzes artificial boundary conditions for random elliptic systems with correlated coefficient fields, enabling accurate finite-domain approximations of the electric field from a localized charge. It extends the Lu-Otto-Wang boundary strategy to correlated media by integrating multiscale logarithmic Sobolev inequalities, two-scale expansions, and multipole corrections (dipole and quadrupole) into a robust framework. A key advance is the construction of sub-linear second-order correctors and the rigorous control of their massive approximations through optimal time-decay estimates of the second-order semigroup, yielding high-probability convergence rates that depend on the domain size $L$, local scale $\ell$, and correlation parameter $\beta$. The results hold for Gaussian-type coefficient fields and more general MSLSI-satisfying media, with numerical simulations corroborating the predicted scaling and illustrating practical performance in correlated random environments.

Abstract

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.

Figures (3)

• Figure 1: Numerical convergence rates of $|\nabla u^{(2L)}(\tfrac{L}{2})- \nabla u^{(L)}(\tfrac{L}{2})|$ for four algorithms with covariance functions: left: $c(|x-y|)=\exp(-\tfrac{|x-y|^2}{8})$; right: $c(|x-y|)=(1+|x-y|/2)^{-5}$.
• Figure 2: Numerical convergence rates with covariance functions: left: $c(|x-y|)=\exp(-\tfrac{|x-y|^2}{16})$; right: $c(|x-y|)=(1+|x-y|/3)^{-5}$.
• Figure 3: Numerical convergence rates with covariance functions: left: $c(|x-y|)=(1+|x-y|)^{-\frac{5}{2}}$; right: $c(|x-y|)=(1+2|x-y|)^{-1.3}$.

Theorems & Definitions (17)

• Corollary 1
• Theorem 1: Effective boundary conditions
• Remark 1
• Proposition 1
• Proposition 2: Large-scale $\mathrm{C}^{0,1}$-estimates for parabolic systems
• Proposition 3: Decay estimates for parabolic systems
• Corollary 2: Decay estimates for elliptic systems
• Theorem 2: Fluctuations of the $2^{\text{nd}}$-order time-dependent flux
• Corollary 3: Optimal time decay of $2^{\text{nd}}$-order semigroups
• Lemma 1: Functional derivative of the $2^{\text{nd}}$-order time dependent flux