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

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