Error estimate and unfolding for periodic homogenization

TL;DR

This paper provides an error estimate for periodic homogenization problems using the periodic unfolding method, quantifying the difference between unfolded gradients and a specific function space without requiring additional regularity assumptions.

Contribution

It introduces a new error bound in periodic homogenization using unfolding, based on the periodic defect of harmonic functions, without extra regularity assumptions.

Findings

Upper bound for the distance between unfolded gradient and target space

Error estimate derived without additional regularity hypotheses

Technical result linking periodic defect to trace norms

Abstract

This paper deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding. We give the upper bound of the distance between the unfolded gradient of a function belonging to $H1(\Omega)$ and the space $\nabla_x H^1(\Omega)\oplus \nabla_y L^2(\Omega ; H^1_{per}(Y))$. These distances are obtained thanks to a technical result presented in Theorem 2.3: the periodic defect of a harmonic function belonging to $H1(Y)$ is written with the help of the norms $H^{1/2}$ of its traces diff erences on the opposite faces of the cell $Y$. The error estimate is obtained without any supplementary hypothesis of regularity on correctors.