Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary
type: Item
instance of: scholarly article
title: Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary
zbMATH Open document ID: 0918.35043
DOI: 10.1007/BF02673049
author: A. Piatnitski
author: Alexander G. Belyaev
author: G. A. Chechkin
published in: Siberian Mathematical Journal
publication date: 1999-05-11T00:00:00Z
review text: 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 \(\varepsilon\).
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35B27
zbMATH DE Number: 1285992
zbMATH Keywords: Poisson equation
zbMATH Keywords: compensated compactness
zbMATH Keywords: interior asymptotic expansion
zbMATH Keywords: boundary layer corrector
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02673049
OpenAlex ID: W2117009991
description: scientific article; zbMATH DE number 1285992