The Bloch Approximation in Periodically Perforated Media

The Bloch Approximation in Periodically Perforated Media We consider a periodically heterogeneous and perforated medium filling an open domain Ω of ℝ N . Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Ω ε (Ω ε being Ω minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where Ωis ℝ N and then localize the problem for a bounded domain Ω, considering a homogeneous Dirichlet condition on the boundary of Ω. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

The Bloch Approximation in Periodically Perforated Media

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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-005-0822-5
Publisher site
See Article on Publisher Site

Abstract

We consider a periodically heterogeneous and perforated medium filling an open domain Ω of ℝ N . Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Ω ε (Ω ε being Ω minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where Ωis ℝ N and then localize the problem for a bounded domain Ω, considering a homogeneous Dirichlet condition on the boundary of Ω.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2005

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