Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T20:32:47.781Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiscale convergence and reiterated homogenisation

Published online by Cambridge University Press:  14 November 2011

G. Allaire  and
M. Briane
G. Allaire
Affiliation:
Commissariat à l'Energie Atomique, DRN/DMT/SERMA, Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette Cedex, France
M. Briane
Affiliation:
Département de Mathématiques, Université Paris 12, 61 ave. du Général de Gaulle, 94040 Créteil Cedex; Laboratoire d'Analyse Numérique, Tour 55-65-5ème étage, Université Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 2 , 1996 , pp. 297 - 342
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Acerbi, E., Piat, V. Chiado, Dal Maso, G. and Percivale, D.. An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18(1992). 481–5.CrossRefGoogle Scholar
2Allaire, G.. Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–518.CrossRefGoogle Scholar
3Allaire, G. and Murat, F.. Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal 7 (1993), 81–95.CrossRefGoogle Scholar
4Amrouche, C. and Girault, V.. Decomposition of vector spaces and application to the Stokesproblem in arbitrary dimension. Czechoslovak Math. J. 44 (1994). 119.CrossRefGoogle Scholar
5Bakhvalov, N. and Panascnko, G.. Homogenization: Averaging Processes in Periodic Media, Mathematics and Its Applications 36 (Dordrecht: Kluwer, 1989).CrossRefGoogle Scholar
6Bensoussan, A. and Lions, J. L.. Homogenizalion with an infinite number of periodic arguments (unpublished notes).Google Scholar
7Bensoussan, A.. Lions, J.L. and Papanicolaou, G.. Asymptotic Analysis for Periodic Structures (Amsterdam: North-Holland. 1978).Google Scholar
8Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Homogenization and ergodic theory. In Stefan Banach Symposium on Probabilities, Banach Center Publications 5, 15–25 (Warsaw; PWN, 1979).Google Scholar
9Bernardi, C.. Contributions à l'analyse numérique de problèmes non linéaires. (These d'Etat, Université Pierre et Marie Curie, 1986).Google Scholar
10Boccardo, L. and Murat, F.. Homogénéisation de problèmes quasi-linéaires. In Atti del Convengno su Sludio dei problemi dell ‘Analisi Funzionale’, Bressanone 79 Sett, 1981, 13–53 (Boloena: Pitagora, 1982).Google Scholar
11Bogovski, M. E.. Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Soviet Math. Dokl. 20 (1979), 1094–8.Google Scholar
12Briane, M.. Corrector for the homogenization of a laminate. Adv. Math. Sci. Appl. 4 (1994), 357– 79.Google Scholar
13Briane, M.. Homogenization of a non-periodic material. J. Math. Pures Appl. 73(1994), 47–66.Google Scholar
14Cabib, E. and Maso, B. Dal. On a class of optimum problems in structural design. J. Opiim. Theory Appl. 56 (1988). 39–65.CrossRefGoogle Scholar
15Cioranescu, D. and Paulin, J. Saint Jean. Homogenization in open sets with holes. J. Math. Anal. Appi. 71 (1979), 590–607.CrossRefGoogle Scholar
16Maso, G. Dal. An Introduction to Gamma-convergence (Boston: Birkhauser, 1993).CrossRefGoogle Scholar
17Dal, G. Maso and Kohn, R.. The local character of G-closer (in prep.).Google Scholar
18Donate, P.Alcune osservazioni sulla convergenza debolc di funzioni non uniformemente oscillanti. Richerche di Matematica 32 (Naples: Univ. Studi Napoli, 1983).Google Scholar
19Donato, P. and Paulin, J. Saint Jean. Homogenization of laplace equation in a porous medium with double periodicity. Japan J. Appl. Math, (to appear).Google Scholar
20Kozlov, S.. Averaging random structures. Soviet Math. Dokl. 19 (1978), 950–4.Google Scholar
21Meyers, N.G.. An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963). 189–206.Google Scholar
22Murat, F.. H-Convergence. In Séminaire d'Analyse Fonctioimelle el Numérique. 1977/78, Univ. d'Alger: English translation to appear in Topics in the mathematical modelling of composite material, ed. Kohn, R. V., Progress in nonlinear differential equations and their equations 1. (Boston: Birkhauser).Google Scholar
23Nguetseng, G.. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 21 (1990), 608–23.Google Scholar
24Papanicolaou, G. and Varadhan, S.. Boundary value problems with rapidly oscillating random coefficients. In Random fields, Esztergom {Hungary 1979), eds Fritz, J. et al. , Colloq. Math. Soc. Janos Bolyai 27 (Amsterdam: North Holland, 1981).Google Scholar
25Sanchez-Palencia, E.. Non-homogeneous media and vibration theory. Lecture Notes in Physics 127 (Berlin: Springer, 1980).Google Scholar
26Spagnolo, S.. Sulla convergenza di soluzioni di cquazioni praboliche e elliliche. Ann. Scuola Norm. Sup. Pisa 22 (1968), 571–97.Google Scholar
27Tartar, L.. Cours Peccol. Collège dc France. 1977 (unpublished, partially written in [22]).Google Scholar
28Tartar, L.. Topics in Nonlinear Analysis (Publications Mathematiques d'Orsay, 1978).Google Scholar
29Zhikov, V., Kozlov, S., Oleinik, O. and Ngoan, N.. Averaging and G-convergence. Russian Math. Surveys 34 (1979), 69–147.CrossRefGoogle Scholar