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

Yves Achdou (1) , Christophe Sabot , Nicoletta Tchou (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 IRMAR - Institut de Recherche Mathématique de Rennes

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. Mathematics Subject Classification. 28A80, 35J05, 35J25, 65N.

Mots clés: domains with fractal boundaries; Helmholtz equation; Neumann boundary conditions; transparent boundary conditons

Domaines: Analyse numérique [math.NA]

Yves Achdou, Christophe Sabot, Nicoletta Tchou. Diffusion and propagation problems in some ramified domains with a fractal boundary. ESAIM: Mathematical Modelling and Numerical Analysis, 2006, 40 (4), pp.623-652. ⟨10.1051/m2an:2006027⟩. ⟨hal-00194810⟩