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Published November 1, 2011 | Published
Journal Article Open

Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated 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 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).

Additional Information

© 2011 Society for Industrial and Applied Mathematics. Received November 08, 2010. Accepted August 03, 2011. Published online November 01, 2011. 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. Finally, we thank Sydney Garstang for proofreading the manuscript. The work of this author was partially supported by the National Science Foundation under award CMMI-092600 and the Department of Energy National Nuclear Security Administration under award DE-FC52-08NA28613. The work of this author was partially supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).

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