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 −∇·(a^ε∇u^ε) = f. On the coefficients a^ε we assume that solutions u^ε of homogeneous ε- problems on simplices with average slope ξ ∈ R^n have the property that flux-averages converge, for ε → 0, to some limit a^∗(ξ), 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 ε-problem in each simplex and are affine on faces