Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast

Abstract

We construct finite-dimensional approximations of solution spaces of divergence-form operators with -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 if source terms are in the unit ball of instead of the unit ball of . Approximation spaces are generated by solving elliptic PDEs on localized subdomains with source terms corresponding to approximation bases for . The -error estimates show that -dimensional spaces with basis elements localized to subdomains of diameter (with ) result in an accuracy for elliptic, parabolic, and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved, provided that localized subdomains contain buffer zones of width , where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).