Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

  • Authors:

  • Yves Achdou;Christophe Sabot;Nicoletta Tchou

  • Affiliations:

  • 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;CNRS, UMPA, UMR 5669, 46, Allee d'Italie, F-69364 Lyon Cedex 07, France;IRMAR, Université de Rennes 1, Rennes, France

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.