Convergent discrete Laplace-Beltrami operators over surfaces

摘要

The convergence problem of the Laplace-Beltrami operators plays an essentialrole in the convergence analysis of the numerical simulations of some importantgeometric partial differential equations which involve the operator. In thisnote we present a new effective and convergent algorithm to compute discreteLaplace-Beltrami operators acting on functions over surfaces. We prove aconvergence theorem for our discretization. To our knowledge, this is the firstconvergent algorithm of discrete Laplace-Beltrami operators over surfaces forfunctions on general surfaces. Our algorithm is conceptually simple and easy tocompute. Indeed, the convergence rate of our new algorithm of discreteLaplace-Beltrami operators over surfaces is $O(r)$ where r represents the sizeof the mesh of discretization of the surface.