The Finite Difference Method, for the heat equation on Sierpi\'{n}ski simplices

TL;DR

This paper develops a finite difference method for solving the heat equation on Sierpiński simplices, providing error analysis, stability conditions, and convergence proofs without relying on eigenvalue approximations.

Contribution

It introduces a novel finite difference scheme for the heat equation on Sierpiński simplices with rigorous theoretical analysis and convergence proof, avoiding eigenvalue approximations.

Findings

The scheme is stable under specified conditions.

The method converges to the true solution.

Error estimates are provided for the scheme.

Abstract

In the sequel, we extend our previous work on the Minkowski Curve to Sierpi\'{n}ski simplices (Gasket and Tetrahedron), in the case of the heat equation. First, we build the finite difference scheme. Then, we give a theoretical study of the error, compute the scheme error, give stability conditions, and prove the convergence of the scheme. Contrary to existing work, we do not call for approximations of the eigenvalues.