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Summary: Transparent Boundary Conditions for the
Helmholtz Equation in Some Ramified
Domains with a Fractal Boundary
Yves Achdou a,b,1
Christophe Sabot c
Nicoletta Tchou d,1
aUFR Math´ematiques, Universit´e Paris 7, Case 7012, 75251 Paris Cedex 05,
France,achdou@math.jussieu.fr
bLaboratoire Jacques-Louis Lions, Universit´e Paris 6, 75252 Paris Cedex 05.
c CNRS, UMPA, UMR 5669, 46, Allee d'Italie, F-69364 Lyon Cedex 07, France,
csabot@umpa.ens-lyon.fr
dIRMAR, Universit´e de Rennes 1, Rennes, France,
nicoletta.tchou@univ-rennes1.fr
Abstract
The paper addresses a class of boundary value problems in some self-similar ram-
ified domains, with the Laplace or Helmholtz equations. Much stress is placed on
transparent boundary conditions which allow the solutions to be computed in sub-
domains. 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
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