A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator

• Published in: Journal of scientific computing Vol. 77; no. 2; pp. 1115 - 1132
• Main Authors: Álvarez, Diego, González-Rodríguez, Pedro, Moscoso, Miguel
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2018; Springer Nature B.V
• Subjects: Algorithms; Approximation; Cardiology; Closed form solutions; Computational Mathematics and Numerical Analysis; Curvature; Exact solutions; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Nodes; Operators (mathematics); Partial differential equations; Radial basis function; Theoretical; Radial basis functions; 65D05; 35R01; 58J05; Cardiology; Surface PDE; Surface Laplacian
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-018-0739-1

In this paper we present a method that uses radial basis functions to approximate the Laplace–Beltrami operator that allows to solve numerically diffusion (and reaction–diffusion) equations on smooth, closed surfaces embedded in R 3 . The novelty of the method is in a closed-form formula for the Laplace–Beltrami operator derived in the paper, which involve the normal vector and the curvature at a set of points on the surface of interest. An advantage of the proposed method is that it does not rely on the explicit knowledge of the surface, which can be simply defined by a set of scattered nodes. In that case, the surface is represented by a level set function from which we can compute the needed normal vectors and the curvature. The formula for the Laplace–Beltrami operator is exact for radial basis functions and it also depends on the first and second derivatives of these functions at the scattered nodes that define the surface. We analyze the converge of the method and we present numerical simulations that show its performance. We include an application that arises in cardiology.