Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical Methods. Part I: Diffusion and Propagation problems.

This paper is devoted to numerical methods for solving boundary value problems in self-similar ramified domains of $\R^2$ with a fractal boundary. Homogeneous Neumann conditions are imposed on the fractal part of the boundary, and Dirichlet conditions are imposed on the remaining part of the boundary. Several partial differential equations are considered. For the Laplace equation, the Dirichlet to Neumann operator is studied. It is shown that it can be computed as the unique fixed point of a rational map. From this observation, a self-similar finite element method is proposed and tested. For the Helmholtz equation, it is shown that the Dirichlet to Neumann operator can also be computed as the limit of an inductive sequence of operators. Here too, a finite element method is designed and tested. It permits to compute numerically the spectrum of the Laplace operator in the irregular domain with Neumann boundary conditions, as well as the eigenmodes. The repartition of the eigenvalues is investigated. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. This permits to solve numerically the wave equation in the self-similar ramified domain