The needle problem approach to non-periodic homogenization

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation $-\nabla\cdot (a^\epsilon\nabla u^\epsilon) = f$. On the coefficients $a^\epsilon$ we assume that solutions $u^\epsilon$ of homogeneous $\epsilon$-problems on simplices with average slope $\xi\in \mathbb{R}^n$ have the property that flux-averages $f a^\epsilon\nabla u^\epsilon\in \mathbb{R}^n$ converge, for $\epsilon\to 0$, to some limit $a^\star(\xi)$, independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an $\epsilon$-problem in each simplex and are affine on faces.