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

Achdou, Yves; Sabot, Christophe; Tchou, Nicoletta

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 623-652.

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.

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