Abstract

CONTRERAS H., LUIS F.  and  GALVIS, JUAN. Finite difference and finite element methods for partial differential equations on fractals. Integración - UIS [online]. 2022, vol.40, n.2, pp.169-190.  Epub May 08, 2023. ISSN 0120-419X.  https://doi.org/10.18273/revint.v40n2-2022003.

In this paper, we present numerical procedures to compute so-lutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian ma-trices and also weak forms of the equation derived using standard length or area measure on a discrete approximation of the fractal set. We then intro-duce 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 con-stant.

        · abstract in Spanish     · text in English     · English ( pdf )