Premium
On the homogenization of some linear problems in domains weakly connected by a system of traps
Author(s) -
Amaziane B.,
Pankratov L.
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.870
Subject(s) - homogenization (climate) , mathematics , bounded function , porous medium , open set , mathematical analysis , domain (mathematical analysis) , parabolic partial differential equation , pure mathematics , porosity , partial differential equation , biodiversity , ecology , geotechnical engineering , engineering , biology
The aim of the paper is to study the asymptotic behaviour of solutions of second‐order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain Ω ε weakly connected by a system of traps ε , where ε is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain Ω ε ⊂Ω a bounded open set of ℝ 3 such that Ω ε =Ω 1 ε ∪Ω 2 ε ∪ ε ∪ W ε , where Ω 1 ε , Ω 2 ε arenon‐intersecting subdomains strongly connected with respect to Ω, ε is a system of traps and meas W ε → 0 as ε → 0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain Ω ε associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology. Copyright © 2007 John Wiley & Sons, Ltd.