A Multiscale Numerical Method for Poisson Problems in Some Ramified Domains with a Fractal Boundary
Yves Achdou, Christophe Sabot, Nicoletta Anna Tchou · Multiscale Modeling and Simulation · 2006
We consider some elliptic boundary value problems in a self‐similar ramified domain of \mathbb{R}^2 with a fractal boundary with Laplace’s equation and nonhomogeneous Neumann boundary conditions. The goal is to approximate the restriction of the solutions to subdomains obtained by stopping the geometric construction after a finite number of steps. For this, we propose a multiscale strategy based on transparent boundary conditions and on a wavelet expansion of the Neumann datum. A self‐similar finite element method is proposed and tested.