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 \(O(H^{-d})\) quadratic (cell) problems on (super-)localized subdomains of size \(O(H \ln^2 H)\); its accuracy (\(O(H)\) in energy norm) is established via the introduction of a new class of higher-order Poincaré inequalities. The method naturally generalizes to time-dependent problems.

This is joint work with Lei Zhang and Leonid Berlyand.