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Published October 19, 2011 | public
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Flux Norm Approach to Homogenization Problems with non-separated Scales


We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L^∞(Ω­), ­Ω ⊂ ℝ^d ) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general case of arbitrary bounded coefficients. For such problems we introduce explicit finite dimensional approximations of solutions with controlled error estimates, which we refer to as homogenization approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical concept of an ∈ family of media as in classical periodic homogenization. We define the flux norm as the L^2 norm of the potential part of the fluxes of solutions, which is equivalent to the usual H^1-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property as the transfer property. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization.

Additional Information

Part of the research of H. Owhadi is supported by the National Nuclear Security Administration through the Predictive Science Academic Alliance Program. The work of L. Berlyand is supported in part by NSF grant DMS-0708324 and DOE grant DE-FG02-08ER25862. We would like to thank L. Zhang for the computations associated with Figure 1. We also thank B. Haines, L. Zhang, and O. Misiats for carefully reading the manuscript and providing useful suggestions. We would like to thank Björn Engquist, Ivo Babuška and John Osborn for useful comments and showing us related and missing references. We are also greatly in debt to Ivo Babuška and John Osborn for carefully reading the manuscript and providing us with very detailed comments and references which have lead to substantial changes.


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