Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary
The aim of this article is to study a model problem for the Poisson equation in a perforated domain with very rapidly oscillating outer boundary in the presence of small dissipation on the boundaries of holes. This model appears in material technology for studying the macroscopic behavior of micro-inhomogeneous perforated media and bodies with rough surfaces. The authors study a particular case of such a medium in which perforation as well as oscillation of the boundary are locally periodic and their structures are assumed to be adjusted. In the study of locally periodic perforation, a difficulty arises: the geometry of cavities is not fixed. The method of compensated compactness or the method of two-scale convergence can be applied for constructing a limit problem, but it provides no estimates for the error. In order to overcome this obstacle, the authors use the technique of asymptotic expansion which requires the data to be regular but makes it possible to estimate the convergence rate. Namely, two terms of the interior asymptotic expansion guarantee an estimate of order \(\sqrt\varepsilon\) in the \(H^1\)-norm. The estimate for the residual is improved by constructing a boundary layer corrector that yields to an estimate of order \(\varepsilon\).
scientific article; zbMATH DE number 1285992
scientific article