High-Order Algorithms for Laplace–Beltrami Operators and Geometric Invariants over Curved Surfaces

Abstract

The Laplace---Beltrami operators on curved surfaces play an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting method with the configuration equation to develop a new effective and convergent generalized finite difference method to approximate the Laplace---Beltrami operators acting on functions over discrete surfaces. The convergence rates of our algorithms of discrete Laplace---Beltrami operators over surfaces is $$O(r^n)$$O(rn), $$n \ge 1$$n?1, where $$r$$r represents the size of the mesh of discretization of the surface. The problem of high-order accuracies will also be discussed and used to compute geometric invariants of the underlying surfaces. Some convergence tests and eigenvalue computations on the sphere, tori and a dumbbell are presented.