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We consider divergence-form scalar elliptic equations and vectorial equationsfor elasticity with rough ($L^\infty(\Omega)$, $\Omega \subset \R^d$)coefficients $a(x)$ that, in particular, model media with non-separated scalesand high contrast in material properties. We define the flux norm as the $L^2$norm of the potential part of the fluxes of solutions, which is equivalent tothe usual $H^1$-norm. We show that in the flux norm, the error associated withapproximating, in a properly defined finite-dimensional space, the set ofsolutions of the aforementioned PDEs with rough coefficients is equal to theerror associated with approximating the set of solutions of the same type ofPDEs with smooth coefficients in a standard space (e.g., piecewise polynomial).We refer to this property as the {\it transfer property}.  A simple application of this property is the construction of finitedimensional approximation spaces with errors independent of the regularity andcontrast of the coefficients and with optimal and explicit convergence rates.This transfer property also provides an alternative to the global harmonicchange of coordinates for the homogenization of elliptic operators that can beextended to elasticity equations. The proofs of these homogenization resultsare based on a new class of elliptic inequalities which play the same role inour approach as the div-curl lemma in classical homogenization.

Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast