A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Ángel García, Mihaela Negreanu, Francisco Ureña and Antonio M. Vargas
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Ángel García: UNED, ETSII, 28040 Madrid, Spain
Mihaela Negreanu: Instituto de Matemática Interdisciplinar, Departamento de Análisis Matemático y Matemática Aplicada, UCM, 28040 Madrid, Spain
Francisco Ureña: UNED, ETSII, 28040 Madrid, Spain
Antonio M. Vargas: Departamento de Análisis Matemático y Matemática Aplicada, UCM, 28040 Madrid, Spain

Mathematics, 2021, vol. 9, issue 22, 1-9

Abstract: The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

Keywords: fractional Laplacian; generalized finite difference method; discrete maximum principle; convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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