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Diffusion and propagation problems in some ramified domains with a fractal boundary

Published online by Cambridge University Press:  15 November 2006

Yves Achdou ,
Christophe Sabot  and
Nicoletta Tchou
Yves Achdou
Affiliation:
UFR Mathématiques, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France and Laboratoire Jacques-Louis Lions, Université Paris 6, 75252 Paris Cedex 05, France. achdou@math.jussieu.fr
Christophe Sabot
Affiliation:
CNRS, UMPA, UMR 5669, 46, Allée d'Italie, 69364 Lyon Cedex 07, France. csabot@umpa.ens-lyon.fr
Nicoletta Tchou
Affiliation:
IRMAR, Université de Rennes 1, Rennes, France. nicoletta.tchou@univ-rennes1.fr
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Abstract

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of ${\mathbb R}^2$ with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous Neumann conditions, the emphasis is placed on transparent boundary conditions, which allow the computation of the solutions in the subdomains obtained by stopping the geometric construction after a finite number of steps. The proposed methods and algorithms will be used numerically in forecoming papers.

Keywords

Domains with fractal boundariesHelmholtz equationNeumann boundary conditionstransparent boundary conditions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 4 , July 2006 , pp. 623 - 652
Copyright
© EDP Sciences, SMAI, 2006

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