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Published October 19, 2011 | Published
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Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast


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 of 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 O(h^−d)-dimensional spaces with basis elements localized to sub-domains of diameter O(h^∞ ln 1/h) (with α ∈ [1/2 , 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 sub-domains contain buffer zones of width O(h^α ln 1/h ) where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

Additional Information

We thank L. Berlyand for stimulating discussions. We also thank Ivo Babuška, John Osborn, George Papanicolaou and Björn Engquist for pointing us in the direction of the localization problem. The work of H. Owhadi is partially supported by the National Science Foundation under Award Number CMMI-092600 and the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613. We thank Sydney Garstang for proofreading the manuscript.

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