Interior error estimate for periodic homogenization

TL;DR

This paper improves the understanding of interior error estimates in periodic homogenization, showing an order of epsilon under certain boundary smoothness conditions and providing global estimates for polygonal domains.

Contribution

It extends previous results by establishing sharper interior error estimates of order epsilon for smooth boundaries and offers global estimates for polygonal and polyhedral domains.

Findings

Interior error is of order epsilon in smooth boundary domains.

Global and interior error estimates are provided for polygonal and polyhedral domains.

Error bounds depend on boundary regularity and domain shape.

Abstract

In a previous article about the homogenization of the classical problem of diff usion in a bounded domain with su ciently smooth boundary we proved that the error is of order $\epsilon^{1/2}$. Now, for an open set with su ciently smooth boundary $C^{1,1}$ and homogeneous Dirichlet or Neuman limits conditions we show that in any open set strongly included in the error is of order $\epsilon$. If the open set $\Omega\subset R^n$ is of polygonal (n=2) or polyhedral (n=3) boundary we also give the global and interrior error estimates.