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

TL;DR

This paper establishes an upper bound for the convergence rate of homogenization in a linear advection-diffusion system with varying diffusivity, validating earlier formal asymptotics through rigorous corrector estimates.

Contribution

It provides the first rigorous corrector estimate for homogenization in media with low and high diffusivity regions, justifying previous formal asymptotic results.

Findings

Proved an upper bound for the convergence rate as  in a heterogeneous medium.

Developed integral estimates for oscillating functions with prescribed averages.

Utilized properties of macroscopic reconstruction operators and energy bounds.

Abstract

We prove an upper bound for the convergence rate of the homogenization limit $\epsilon\to 0$ for a linear transmission problem for a advection-diffusion(-reaction) system posed in areas with low and high diffusivity, where $\epsilon$ is a suitable scale parameter. On this way, we justify the formal homogenization asymptotics obtained by us earlier by proving an upper bound for the convergence rate (a corrector estimate). The main ingredients of the proof of the corrector estimate 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.