 
Summary: A MULTISCALE NUMERICAL METHOD FOR POISSON
PROBLEMS IN SOME RAMIFIED DOMAINS WITH A FRACTAL
BOUNDARY
YVES ACHDOU , CHRISTOPHE SABOT , AND NICOLETTA TCHOU
Abstract. We consider some elliptic boundary value problems in a selfsimilar ramified
domain of R2 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 selfsimilar finite element method is proposed and tested.
1. Introduction. This paper is concerned with some boundary value prob
lems in a selfsimilar ramified domain of R2
with a fractal boundary. It was inspired
by a wider and challenging project aimed at simulating the diffusion of medical
sprays in lungs. Our goal are more modest here, since the geometry of the problems
(only two dimensions) and the physical phenomena considered are much simpler,
but we hope that rigorous results and methods will prove useful.
The domain called 0
, is displayed on Figure 2.1. It is constructed in an infinite
number of steps, starting from a simple polygonal Tshaped domain of R2
