EU Regional School - Muntean Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Dr. Muntean - The Homogenization Method and Multiscale Modeling

Institute for Complex Molecular Systems (ICMS)
Eindhoven University of Technology

Abstract

The lecture starts with the study of an oscillatory elliptic PDE formulated rstly in xed and, afterwards, in periodically-perforated domains. We remove the oscillations by means of a (formal) asymptotic homogenization method. The output of this procedure consists of a \guessed" averaged model equations and explicit rules (based on cell problems) for computing the e ective coecients. About half of the lecture will be spent on explaining the basic steps of the averaging procedure and the way this can be put in action when other PDE problems need to be tackled.
In the second part of the lectures, I introduce the concept of two-scale conver- gence (and correspondingly, the two-scale compactness) in the sense of Allaire and Nguetseng and derive rigorously the averaged PDE models and coecients obtained previously. [This step uses the framework of Sobolev and Bochner spaces and relies on basic tools like weak convergence methods, compact em- beddings as well as extension theorems in Sobolev spaces. I will be very brief on these aspects, but suciently clear such that a large audience, which is not necessarily a mathematical one, can follow.]
I will particularly emphasize the key aspect { the role the choice of mi- crostructures (pores, perforations, subgrids, etc.) plays in performing the overall averaging procedure.

Basic Ref: A. Muntean, V. Chalupecky, Homogenization Method and Multiscale Modeling, Lecture notes at the Institute for Mathematics and Industry, Kyushu University, 2011. 

Lecture Material I
Lecture Material II