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

Yves Achdou; Christophe Sabot; Nicoletta Tchou

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 4, page 623-652
  • ISSN: 0764-583X

Abstract

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This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of 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.

How to cite

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Achdou, Yves, Sabot, Christophe, and Tchou, Nicoletta. "Diffusion and propagation problems in some ramified domains with a fractal boundary." ESAIM: Mathematical Modelling and Numerical Analysis 40.4 (2006): 623-652. <http://eudml.org/doc/249734>.

@article{Achdou2006,
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. },
author = {Achdou, Yves, Sabot, Christophe, Tchou, Nicoletta},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Domains with fractal boundaries; Helmholtz equation; Neumann boundary conditions; transparent boundary conditions.; elliptic boundary value problems; Neumann and transparent boundary conditions; Poisson equation; Laplace equation},
language = {eng},
month = {11},
number = {4},
pages = {623-652},
publisher = {EDP Sciences},
title = {Diffusion and propagation problems in some ramified domains with a fractal boundary},
url = {http://eudml.org/doc/249734},
volume = {40},
year = {2006},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2006/11//
PB - EDP Sciences
VL - 40
IS - 4
SP - 623
EP - 652
AB - 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.
LA - eng
KW - Domains with fractal boundaries; Helmholtz equation; Neumann boundary conditions; transparent boundary conditions.; elliptic boundary value problems; Neumann and transparent boundary conditions; Poisson equation; Laplace equation
UR - http://eudml.org/doc/249734
ER -

References

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