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Revista Integración
Print version ISSN 0120-419XOn-line version ISSN 2145-8472
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.
MSC2010:
65N30, 28A80, 35J20, 35J15.
Keywords : Fractal diffusion; Laplacian on a fractal; Renormalization con-stant.