Asymptotic Analysis of Boundary Layer Correctors and Applications

TL;DR

This paper develops second order boundary layer correctors for homogenization problems, achieving optimal error estimates in nonsmooth settings and extending convergence results for multiscale finite element methods.

Contribution

It introduces suitable second order boundary layer correctors for homogenization, improving error estimates in nonsmooth coefficient cases and extending convergence proofs for multiscale methods.

Findings

Achieves $O(\epsilon^{3/2})$ error estimate in 2D with nonsmooth coefficients.

Extends error estimates to 3D with certain regularity assumptions.

Links homogenization error analysis to multiscale finite element method convergence.

Abstract

In this paper we extend the ideas presented in Onofrei and Vernescu [\textit{Asymptotic Analysis, 54, 2007, 103-123}] and introduce suitable second order boundary layer correctors, to study the $H^1$-norm error estimate for the classical problem in homogenization. Previous second order boundary layer results assume either smooth enough coefficients (which is equivalent to assuming smooth enough correctors $\chi_j,\chi_{ij}\in W^{1,\infty}$), or smooth homogenized solution $u_0$, to obtain an estimate of order $\displaystyle O(\epsilon^{\frac{3}{2}})$. For this we use the periodic unfolding method developed by Cioranescu, Damlamian and Griso [\textit{C. R. Acad. Sci. Paris, Ser. I 335, 2002, 99-104}]. We prove that in two dimensions, for nonsmooth coefficients and general data, one obtains an estimate of order $\displaystyle O(\epsilon^\frac{3}{2})$. In three dimenssions the same estimate is obtained assuming $\chi_j,\chi_{ij}\in W^{1,p}$, with $p>3$. We also discuss how our results extend, in the case of nonsmooth coefficients, the convergence proof for the finite element multiscale method proposed by T.Hou et al. [\textit{ J. of Comp. Phys., 134, 1997, 169-189}] and the first order correctoranalysis for the first eigenvalue of a composite media obtained by Vogelius et al.[\textit{Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299}].