The finite volume method on Sierpi\'nski simplices

TL;DR

This paper develops a finite volume method for the Laplacian on Sierpiński gaskets using Strichartz average, demonstrating stability and convergence similar to finite difference methods.

Contribution

It introduces a novel finite volume approach on fractals based on Strichartz average, expanding numerical methods for fractal Laplacians.

Findings

Method shows stability comparable to finite difference methods.

Convergence of the method is established.

Provides a new tool for numerical analysis on fractals.

Abstract

In this work, we exploit Strichartz average approach \cite{Strichartz2001} to define the Laplacian on Sierpi\'{n}ski gasket, in the construction of the finite volume method. The approach present sum similarities with the finite difference approach in terms of stability and convergence.