Convergent discrete Laplace-Beltrami operators over surfaces

The convergence problem of the Laplace-Beltrami operators plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve the operator. In this note we present a new effective and convergent algorithm to compute discrete Laplace-Beltrami operators acting on functions over surfaces. We prove a convergence theorem for our discretization. To our knowledge, this is the first convergent algorithm of discrete LaplaceBeltrami operators over surfaces for functions on general surfaces. Our algorithm is conceptually simple and easy to compute. Indeed, the convergence rate of our new algorithm of discrete Laplace-Beltrami operators over surfaces is O(r) where r represents the size of the mesh of discretization of the surface.