Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Multiscale convergence and reiterated homogenisation

G. Allairea1 and M. Brianea2a3

a1 Commissariat à l'Energie Atomique, DRN/DMT/SERMA, Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette Cedex, France

a2 Département de Mathématiques, Université Paris 12, 61 ave. du Général de Gaulle, 94040 Créteil Cedex;

a3 Laboratoire d'Analyse Numérique, Tour 55-65-5ème étage, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France

This paper generalises the notion of two-scale convergence to the case of multiple separated scales of periodic oscillations. It allows us to introduce a multi-scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients. This new method is applied to a model problem with a finite or infinite number of microscopic scales, namely the homogenisation of the heat equation in a composite material. Finally, it is generalised to handle the homogenisation of the Neumann problem in a perforated domain.

(Received June 29 1994)