Mesoscale Asymptotic Approximations to Solutions of Mixed Boundary Value Problems in Perforated Domains

We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the Neumann boundary conditions being prescribed on the surfaces of small voids. The only assumption made on...

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Bibliographic Details
Published in:Multiscale modeling & simulation 2011-01, Vol.9 (1), p.424-448
Main Authors: Maz'ya, V, Movchan, A, Nieves, M
Format: Article
Language:English
Subjects:
Approximation
Asymptotic methods
Asymptotic properties
Boundary value problems
Estimates
Mathematical analysis
Mathematical models
MEDICIN
MEDICINE
mesoscale approximation
multiply perforated domain
Perforations
Poisson distribution
singular perturbation
Studies
Voids
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Summary:We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the Neumann boundary conditions being prescribed on the surfaces of small voids. The only assumption made on the geometry is that the diameter of a void is assumed to be smaller compared to the distance to the nearest neighbor. The asymptotic approximation, obtained here, involves a linear combination of dipole fields constructed for individual voids, with the coefficients, which are determined by solving a linear algebraic system. We prove the solvability of this system and derive an estimate for its solution. The energy estimate is obtained for the remainder term of the asymptotic approximation. [PUBLICATION ABSTRACT]
ISSN:1540-3459
1540-3467
DOI:10.1137/100791294