Finite difference and finite element methods for partial differential equations on fractals

Abstract

In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or área measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree.

Keywords

• Fractal diffusion
• Laplacian on a fractal
• Renormalization constant