Asymptotic Analysis of Boundary Layer Correctors and Applications

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 $χ_j,χ_{ij}\in W^{1,\infty}$), or smooth homogenized solution $u_0$, to obtain an estimate of order $\displaystyle O(ε^{\frac{3}{2}})$. 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 $\displaystyle O(ε^\frac{3}{2})$. In three dimenssions the same estimate is obtained assuming $χ_j,χ_{ij}\in W^{1,p}$, 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}].