This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of 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.
Mots-clés : domains with fractal boundaries, Helmholtz equation, Neumann boundary conditions, transparent boundary conditions
@article{M2AN_2006__40_4_623_0,
author = {Achdou, Yves and Sabot, Christophe and Tchou, Nicoletta},
title = {Diffusion and propagation problems in some ramified domains with a fractal boundary},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {623--652},
publisher = {EDP-Sciences},
volume = {40},
number = {4},
year = {2006},
doi = {10.1051/m2an:2006027},
mrnumber = {2274772},
zbl = {1112.65115},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006027/}
}
TY - JOUR AU - Achdou, Yves AU - Sabot, Christophe AU - Tchou, Nicoletta TI - Diffusion and propagation problems in some ramified domains with a fractal boundary JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 623 EP - 652 VL - 40 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006027/ DO - 10.1051/m2an:2006027 LA - en ID - M2AN_2006__40_4_623_0 ER -
%0 Journal Article %A Achdou, Yves %A Sabot, Christophe %A Tchou, Nicoletta %T Diffusion and propagation problems in some ramified domains with a fractal boundary %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 623-652 %V 40 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006027/ %R 10.1051/m2an:2006027 %G en %F M2AN_2006__40_4_623_0
Achdou, Yves; Sabot, Christophe; Tchou, Nicoletta. Diffusion and propagation problems in some ramified domains with a fractal boundary. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 623-652. doi : 10.1051/m2an:2006027. http://www.numdam.org/articles/10.1051/m2an:2006027/
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