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A Local Radial Basis Function Method for the Laplace–Beltrami Operator

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Abstract

We introduce a new local meshfree method for the approximation of the Laplace–Beltrami operator on a smooth surface in \({\mathbb {R}}^3\). It is a direct method that uses radial basis functions augmented with multivariate polynomials. A key element of this method is that it does not need an explicit expression of the surface, which can be simply defined by a set of scattered nodes. Likewise, it does not require expressions for the surface normal vectors or for the curvature of the surface, which are approximated using explicit formulas derived in the paper. An additional advantage is that it is a local method and, hence, the matrix that approximates the Laplace–Beltrami operator is sparse, which translates into good scalability properties. The convergence, accuracy and other computational characteristics of the proposed method are studied numerically. Its performance is shown by solving two reaction–diffusion partial differential equations on surfaces; the Turing model for pattern formation, and the Schaeffer’s model for electrical cardiac tissue behavior.

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Acknowledgements

This work has been supported by Spanish MICINN Grant FIS2016-77892-R.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain

    Diego Álvarez, Pedro González-Rodríguez & Manuel Kindelan

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  1. Diego Álvarez
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  2. Pedro González-Rodríguez
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  3. Manuel Kindelan
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Correspondence to Diego Álvarez.

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Álvarez, D., González-Rodríguez, P. & Kindelan, M. A Local Radial Basis Function Method for the Laplace–Beltrami Operator. J Sci Comput 86, 28 (2021). https://doi.org/10.1007/s10915-020-01399-3

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  • DOI: https://doi.org/10.1007/s10915-020-01399-3

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