# Local precised approximation in multiscale problems with local defects

## TL;DR

This paper demonstrates that in multiscale diffusion problems with localized defects, the corrector function can effectively approximate solutions with accuracy comparable to the purely periodic case, with convergence rates depending on defect integrability.

## Contribution

It extends homogenization approximation techniques to multiscale problems with localized defects, providing explicit convergence rates based on defect integrability.

## Key findings

- Corrector functions approximate solutions effectively.
- Convergence rates depend on defect's $L^r$ integrability.
- Extension of periodic homogenization methods to defective media.

## Abstract

We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r (R^d ) , 1 < r < +$\infty$$ , function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $L^r$ integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [13], and borrows a lot from it. The details of the results announced in this Note are given in our forthcoming publications [2,12].