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 L^∞-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 subdomains with source terms corresponding to approximation bases for H^2. The H^1-error estimates show that O(h^(−d))-dimensional spaces with basis elements localized to subdomains of diameter O(hα ln math) (with α ∊ [½,1)) result in an O(h^(2−2α)) 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 O(h^α ln 1/4), where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

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