Comptes Rendus
no. 10
Partial Differential Equations
Matching and multiscale expansions for a model singular perturbation problem
[Développements raccordé et multi-échelle pour un problème de perturbation singulière modèle]

Présenté par : Alain Bensoussan

Sébastien Tordeux1 ; Grégory Vial2 ; Monique Dauge3
1 MIP, INSA Toulouse, 31077 Toulouse, France
2 IRMAR, ENS Cachan Bretagne, 35170 Bruz, France
3 IRMAR, Université de Rennes 1, 35042 Rennes, France
Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 637-642.

On considère le problème de Laplace–Dirichlet dans un domaine polygonal qui présente une perturbation de taille ε en l'un de ses sommets. Cette perturbation est supposée auto-similaire, i.e. provient d'un motif fixe dilaté à l'échelle ε. Sur ce problème modèle, nous mettons en œuvre deux méthodes : développements asymptotiques raccordés et développement multi-échelle. Nous mettons en évidence les particularités de chaque approche et montrons comment passer d'un développement à l'autre.

We consider the Laplace–Dirichlet equation in a polygonal domain which is perturbed at the scale ε near one of its vertices. We assume that this perturbation is self-similar, that is, derives from the same pattern for all values of ε. On the base of this model problem, we compare two different approaches: the method of matched asymptotic expansions and the method of multiscale expansion. We enlighten the specificities of both techniques, and show how to switch from one expansion to the other.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.10.010
Sébastien Tordeux 1 ; Grégory Vial 2 ; Monique Dauge 3

1 MIP, INSA Toulouse, 31077 Toulouse, France
2 IRMAR, ENS Cachan Bretagne, 35170 Bruz, France
3 IRMAR, Université de Rennes 1, 35042 Rennes, France 
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     title = {Matching and multiscale expansions for a model singular perturbation problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {637--642},
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     volume = {343},
     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.10.010},
     language = {en},
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Sébastien Tordeux; Grégory Vial; Monique Dauge. Matching and multiscale expansions for a model singular perturbation problem. Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 637-642. doi : 10.1016/j.crma.2006.10.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.010/

[1] G. Caloz, M. Costabel, M. Dauge, G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptotic Analysis (2006), in press

[2] M. Dauge Elliptic Boundary Value Problems in Corner Domains – Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988

[3] P. Grisvard Boundary Value Problems in Non-Smooth Domains, Pitman, London, 1985

[4] A. Il'lin Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, 1992

[5] P. Joly; S. Tordeux Matching of asymptotic expansions for wave propagation in media with thin slots I: The asymptotic expansion, Multiscale Modeling and Simulation, Volume 5 (2006) no. 1, pp. 304-336

[6] V.A. Kondrat'ev Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., Volume 16 (1967), pp. 227-313

[7] V. Maz'ya; S.A. Nazarov; B.A. Plamenevskij Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Birkhäuser, Berlin, 2000

[8] S. Tordeux, G. Vial, M. Dauge, Selfsimilar perturbation near a corner: matching and multiscale expansions, 2006, in preparation

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