Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

Abstract

We prove an upper bound for the convergence rate of the homogenization limit ε → 0 for a linear transmission problem for a advection–diffusion(–reaction) system posed in areas with low and high diffusivity, where ε is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity.Eur. J. Appl. Math.22, 493–516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.