Flux Norm Approach to Homogenization Problems with Non-separated Scales

We consider 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 ǫ 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.