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

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.