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 approximatethe Laplace&-Beltrami operator that allows to solve numerically diffusion (and reaction&-diffusion) equations on smooth, closed surfaces embedded in R3. The novelty of the methodis in a closed-form formula for the Laplace&-Beltrami operator derived in the paper, whichinvolve 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 knowledgeof the surface, which can be simply defined by a set of scattered nodes. In that case, thesurface is represented by a level set function from which we can compute the needed normalvectors and the curvature. The formula for the Laplace&-Beltrami operator is exact for radialbasis functions and it also depends on the first and second derivatives of these functionsat the scattered nodes that define the surface. We analyze the converge of the method andwe present numerical simulations that show its performance. We include an application thatarises in cardiology.

A closed-form formula for the RBF-based approximation of the Laplace-Beltrami operator Articles