Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary

Yves Achdou; Nicoletta Tchou

doi:10.1016/j.crma.2006.02.025

Numerical Analysis

Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary

Presented by: Olivier Pironneau

Yves Achdou ^1,²; Nicoletta Tchou ³

¹ UFR mathématiques, université Paris 7, case 7012, 75251 Paris cedex 05, France
² Laboratoire Jacques-Louis Lions, université Paris 6, 75252 Paris cedex 05, France
³ IRMAR, université de Rennes 1, 35042 Rennes cedex, France

Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 611-616.

Abstract
Résumé

This note deals with some boundary value problems in a self-similar ramified domain of $R^{2}$ , with a fractal boundary. The partial differential equation is Laplace's equation, and there are nonhomogeneous generalized Neumann boundary conditions on the fractal boundary. We propose a multiscale strategy for approximating the restriction of the solutions to simple subdomains. This strategy is based on transparent boundary conditions and on a wavelet expansion of the Neumann datum.

On considère une classe de problèmes aux limites dans un domaine autosimilaire ramifié de $R^{2}$ avec une frontière fractale. L'équation aux dérivées partielles est l'équation de Laplace et on impose des conditions de Neumann non homogènes sur la frontière fractale. On propose une stratégie multiéchelle pour calculer la solution dans des sous-domaines simples. Cette stratégie met en jeu un développement de la donnée de Neumann sur une base d'ondelettes et des conditions aux limites transparentes.

Received: 2005-09-05
Accepted: 2006-02-22
Published online: 2006-03-24

DOI: 10.1016/j.crma.2006.02.025

Author's affiliations:

Yves Achdou ^{1, 2}; Nicoletta Tchou ³

¹ UFR mathématiques, université Paris 7, case 7012, 75251 Paris cedex 05, France
² Laboratoire Jacques-Louis Lions, université Paris 6, 75252 Paris cedex 05, France
³ IRMAR, université de Rennes 1, 35042 Rennes cedex, France

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     author = {Yves Achdou and Nicoletta Tchou},
     title = {Boundary value problems with nonhomogeneous {Neumann} conditions on a fractal boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {611--616},
     publisher = {Elsevier},
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     number = {8},
     year = {2006},
     doi = {10.1016/j.crma.2006.02.025},
     language = {en},
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Yves Achdou; Nicoletta Tchou. Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 611-616. doi : 10.1016/j.crma.2006.02.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.025/

References
Cited by

[1] Y. Achdou, C. Sabot, N. Tchou, A multiscale numerical method for Poisson problems in some ramified domains with a fractal boundary, submitted for publication

[2] Y. Achdou; C. Sabot; N. Tchou Transparent boundary conditions for a class of boundary values problems in some ramified domains with a fractal boundary, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006) (in this issue)

[3] A. Jonsson; H. Wallin Function spaces on subsets of $R^{n}$ , Math. Rep., Volume 2 (1984) no. 1, pp. 1-221

[4] V.G. Maz'ja Sobolev Spaces, Springer Ser. in Soviet Math., Springer-Verlag, Berlin, 1985 (Translated from the Russian by T.O. Shaposhnikova)

Cited by Sources:

^⁎ The research of Yves Achdou and Nicoletta Tchou was partially supported by the ACINIM “LePoumonVousDisJe”. Le travail de Yves Achdou et de Nicoletta Tchou a été partiellement supporté par l'ACINIM “LePoumonVousDisJe”.

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