Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast
Houman Owhadi, Lei Zhang · Multiscale Modeling and Simulation · 2011
We construct finite-dimensional approximations of solution spaces of divergence-form operators with [Formula: see text]-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 [Formula: see text] if source terms are in the unit ball of [Formula: see text] instead of the unit ball of [Formula: see text]. Approximation spaces are generated by solving elliptic PDEs on localized subdomains with source terms corresponding to approximation bases for [Formula: see text]. The [Formula: see text]-error estimates show that [Formula: see text]-dimensional spaces with basis elements localized to subdomains of diameter [Formula: see text] (with [Formula: see text]) result in an [Formula: see text] 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 [Formula: see text], where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).