 
Summary: Neumann Conditions on Fractal Boundaries
Yves Achdou
, Nicoletta Tchou
.
June 18, 2007
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 Hausdorff dimension of the fractal boundary is greater than one. The goal
is twofold: first rigorously define the boundary value problems, second approximate the
restriction of the solutions to subdomains obtained by stopping the geometric construction
after a finite number of steps. For the first task, a key step is the definition of a trace operator.
For the second task, a multiscale strategy based on transparent boundary conditions and on
a wavelet expansion of the Neumann datum is proposed, following an idea contained in a
previous work by the same authors. Error estimates are given and numerical results are
presented.
selfsimilar domain, fractal boundary, partial differential equations
35J25, 35J05, 28A80, 65N
1 Introduction
This work is concerned with some boundary value problems in some selfsimilar ramified domains
