| A new approach to homogenization with arbitrarily rough and high contrast coefficients for scalar and vectorial problems (2009) | |||||||||
Abstract | |||||||||
| We consider divergence-form (elliptic, parabolic and hyperbolic) equations (or systems of equations for elasticity) with rough ($L^\infty(\Omega)$, $\Omega \subset \R^d$) coefficients that, in particular, may contain infinitely many non-separated scales. The homogenization of these equations with periodic or ergodic coefficients and well separated scales is now well understood. In this work, for the most general case of arbitrary bounded coefficients, we construct explicit finite dimensional (homogenization) approximations of solutions with controlled error estimates. In particular, our approach allows one to analyze a given medium directly without introducing the mathematical concept of an $\epsilon$ family of media. We also obtain an explicit error constant which is independent of the contrast of the material and geometry of its microstructure. Next, we minimize the number of pre-computed problems for homogenization with arbitrary bounded coefficients by introducing a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization. Finally, we provide an example of localization in the pre-computation. Our approach is a generalization of a method introduced by Babuska Caloz and Osborn for laminar media. The generalization is based on preservation of fluxes across coarse sub-domains.. Comment: 63 pages | |||||||||
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