Neumann conditions on fractal boundaries

Article type: Research Article

Authors: Achdou, Yves | Tchou, Nicoletta

Affiliations: UFR Mathématiques, Université Paris Diderot, Case 7012, 75251 Paris Cedex 05, France and Laboratoire Jacques-Louis Lions, Université Paris 6, 75252 Paris Cedex 05 E-mail: achdou@math.jussieu.fr | IRMAR, Université de Rennes 1, Rennes, France E-mail: nicoletta.tchou@univ-rennes1.fr

Abstract: 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 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.

Keywords: self-similar domain, fractal boundary, partial differential equations

Journal: Asymptotic Analysis, vol. 53, no. 1-2, pp. 61-82, 2007