The needle problem approach to non-periodic homogenization

Abstract

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation -∇·(aε∇uε) = f. On the coefficients aε we assume that solutions uε of homogeneous ε- problems on simplices with average slope ξ ∈ R{double struck}n have the property that flux- CMEX8.-1.integraltext εεn*averages - a ∇u ∈ R 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. © American Institute of Mathematical Sciences.