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Center for Nonlinear Studies |
Classical homogenization theory deals with mathematical models of strongly inhomogeneous media described by PDEs with rapidly oscillating coefficients of the form A(x/ε), ε → 0. The goal is to approximate this problem by a homogenized (simpler) PDE with slowly varying coefficients that do not depend on the small parameter ε. The original problem has two scales: fine O(ε) and coarse O(1), whereas the homogenized problem has only a coarse scale. The homogenization of PDEs with periodic or random ergodic coefficients and well-separated scales is well understood. In a joint work with H. Owhadi (Caltech) we consider the most general case of arbitrary L∞ coefficients, which may contain infinitely many scales that are not necessarily well-separated. Specifically, we study scalar and vectorial divergence-form elliptic PDEs with such coefficients. We es- tablish two finite-dimensional homogenization approximations that generalize the correctors in classical homogenization. We introduce a flux norm and establish the error estimate in this norm with an explicit and optimal error constant independent of the contrast and regularity of the coefficients. A proper generalization of the notion of a cell problem is the key issue in our consideration. If time permits, we will discuss most recent results (L. Zhang, Owhadi) on lo- calized multiscale basis that allows for numerical implementation of our theoretical results and work in progress (with Fedorov, Owhadi and Zhang) on self-assembled protein aggregates.
Host: Vitaliy Gyrya, T-5, 5-2729