Homogenization with arbitrary rough coefficients and super-localization

Houman Owhadi
California Institute of Technology

Abstract: We introduce a new method for the numerical homogenization of divergence form elliptic equations with arbitrary rough ($\L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization.
The approximation space is generated as an interpolation space (over a coarse mesh of resolution $H$) minimizing the $\L^2$ norm of source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H (\ln H)^2)$; its accuracy ($\mathcal{O}(H)$ in energy norm) is established via the introduction of a new class of higher-order Poincar\'{e} inequalities. The method naturally generalizes to time dependent problems. This is a joint work with Lei Zhang and Leonid Berlyand.

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