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

A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator 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 \$\$\mathbb {R}^3\$\$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

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

, Volume 77 (2) – May 29, 2018
18 pages

/lp/springer_journal/a-closed-form-formula-for-the-rbf-based-approximation-of-the-laplace-M3PTw050vK
Publisher
Springer Journals
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
DOI
10.1007/s10915-018-0739-1
Publisher site
See Article on Publisher Site

### Abstract

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 \$\$\mathbb {R}^3\$\$ 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.

### Journal

Journal of Scientific ComputingSpringer Journals

Published: May 29, 2018