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

TL;DR

This paper develops numerical methods using finite difference and finite element techniques to solve partial differential equations on fractal sets, including procedures for normalization and specific case studies like the Sierpinski triangle.

Contribution

It introduces a novel numerical framework for PDEs on fractals, including normalization procedures and applications to complex fractal geometries.

Findings

Effective numerical procedures for PDEs on fractals.

Successful application to Sierpinski triangle and Hata tree.

Demonstrated normalization of diffusions 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 area 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.