Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast

TL;DR

This paper introduces a new finite-dimensional approximation method for divergence form operators with high-contrast and non-separated scales, avoiding ergodicity and scale separation assumptions, and achieves high accuracy through localized basis functions.

Contribution

The authors develop a localized basis construction method for homogenization that does not rely on traditional scale separation or ergodicity, suitable for high-contrast media.

Findings

Achieves $ ext{O}(h^{2-2eta})$ accuracy with basis functions localized to small sub-domains.

Method preserves accuracy in high-contrast media with buffer zones of bounded contrast.

Applicable to elliptic, parabolic, hyperbolic, and vectorial PDEs.

Abstract

We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^\alpha \ln \frac{1}{h})$ (with $\alpha \in [1/2,1)$) result in an $\mathcal{O}(h^{2-2\alpha})$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^\alpha \ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).