# Correctors and error estimates for reaction-diffusion processes through thin heterogeneous layers in case of homogenized equations with interface diffusion

## TL;DR

This paper develops and validates corrector-based approximations for reaction-diffusion equations in domains with thin heterogeneous layers, providing error estimates that quantify the approximation accuracy as the layer thickness tends to zero.

## Contribution

The paper introduces a method to construct accurate approximations of microscopic solutions using correctors for reaction-diffusion problems with thin heterogeneous layers, along with rigorous error estimates.

## Key findings

- Error estimates of order ^{1/2} in H^1-norm for the approximations.
- Effective interface conditions derived in the homogenization limit.
- Validation of the approximation accuracy through rigorous error bounds.

## Abstract

In this paper, we construct approximations of the microscopic solution of a nonlinear reaction--diffusion equation in a domain consisting of two bulk-domains, which are separated by a thin layer with a periodic heterogeneous structure. The size of the heterogeneities and thickness of the layer are of order $\epsilon$, where the parameter $\epsilon$ is small compared to the length scale of the whole domain. In the limit $\epsilon \to 0$, when the thin layer reduces to an interface $\Sigma$ separating two bulk domains, a macroscopic model with effective interface conditions across $\Sigma$ is obtained. Our approximations are obtained by adding corrector terms to the macroscopic solution, which take into account the oscillations in the thin layer and the coupling conditions between the layer and the bulk domains. To validate these approximations, we prove error estimates with respect to $\epsilon$. Our approximations are constructed in two steps leading to error estimates of order $\epsilon^{\frac12}$ and $\epsilon$ in the $H^1$-norm.