Naslov
Small-amplitude homogenisation of diffusion equation

Autori
Antonić, Nenad ; Vrdoljak, Marko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Fifth International Conference of Applied Mathematics and Computing, volume 1 / Nenov, Svetoslav - Plovdiv, 2008, 39-39

Skup
Fifth International Conference of Applied Mathematics and Computing

Mjesto i datum
Plovdiv, Bugarska, 12.08.2008. - 18.08.2008

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
H-convergence ; small-amplitude homogenisation ; diffusion equation ; H-measure

Sažetak
Abstract theory of non-periodic homogenisation for non-stationary diffusion equation is much less known than the corresponding theory for stationary diffusion. However, the main results of G-convergence and H-convergence were already obtained by Sergio Spagnolo in the seventies, with some extensions by Vasilij V. Žikov and collaborators in the eighties, and Andrea Dell'Aglio and Francois Murat in the nineties. We shall review these results in a systematic manner, following the approach of Luc Tartar in the elliptic case. Besides, we shall prove that the smoothness (with respect to a parameter) is preserved in the process of taking the H-limit, which is essential for our purposes. The small-amplitude homogenisation consists in taking a sequence of coefficients which difference is proportional to a small parameter, and then computing the first correction in the limit. The explicit formula for the correction in the elliptic case can in general be obtained by using H-measures, a tool introduced arround 1990 by Luc Tartar and Patrick Gerard. For parabolic problems, those classical H-measures are not well suited. Recently, the first author (jointly with Martin Lazar) introduced several parabolic variants, which allowed a number of applications to be extended from elliptic to parabolic equations. By using such a variant of H-measures we were able to write the explicit expression for the correction in the parabolic case. Under periodic assumption on the coefficients, the effective coefficients can be computed explicitly by classical methods. This gives us an opportunity to compare the results obtained by the two methods.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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