W. G. Pritchard Lab Seminar: 3:30-4:30 PM, 116 McAllister Building
**Monday February 24, 2003**
Asymptotic partial domain decomposition and partial homogenization
Grigori Panasenko
Equipe d'Analyse Numerique
Universite Jean Monnet de St. Etienne
Abstract:
The method of partial asymptotic domain decomposition was recently proposed
for partial differential equations, formulated for rod structures, i.e. in
some connected unions of thin cylinders. It is based on the information
about the structure of the asymptotic solution in different parts of such
complicated domain. The principal idea of the method is to extract the
subdomain of singular behaviour of the solution and to reduce dimension of
the problem in the subdomain of regular behaviour of the solution. The
special interface conditions are imposed on the common boundary of these
partially decomposed subdomains. This method can be realized numerically as
a super-element version of the finite element method.
We apply this method to the classical homogenization problem simulating
physical fields in composite materials. Homogenization techniques are an
effective tool for simulation of macroscopic behaviour of
micro-heterogeneous media. Nevertheless, there is a difficulty in the
analysis of boundary layers, so asymptotic higher order approximations are
constructed only in some particular cases. We propose and justify here the
partial homogenization that keeps the initial equation in some thin
subdomain, homogenizes the equation in the remaining part of the domain,
and prescribes the appropriate interface conditions for homogenized and
non-homogenized parts. This approach avoids the construction of boundary
layers.