The Finite Difference Method, for the heat equation on Sierpiński simplices
It offers a rigorous numerical method for solving PDEs on fractal domains, which is a niche problem for mathematicians and physicists studying fractal geometry.
The paper extends the finite difference method to solve the heat equation on Sierpiński simplices, providing error analysis, stability conditions, and convergence proof without eigenvalue approximations.
In the sequel, we extend our previous work on the Minkowski Curve to Sierpiński 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.