Asymptotic Analysis of Boundary Layer Correctors and Applications

Original · EN

In this paper we extend the ideas presented in Onofrei and Vernescu [Asymptotic Analysis, 54, 2007, 103-123] and introduce suitable second order boundary layer correctors, to study the H¹-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 χⱼ,χij∈ W¹,∞), or smooth homogenized solution u₀, to obtain an estimate of order O(ε³/²). For this we use the periodic unfolding method developed by Cioranescu, Damlamian and Griso [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 O(ε³/2). In three dimenssions the same estimate is obtained assuming χⱼ,χij∈ W¹,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. [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.[Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299].