Convergence of Discrete Laplace-Beltrami Operators over Surfaces

The convergence property of the discrete Laplace-Beltrami operator is the foundation of convergence analysis of the numerical simulation process of some geometric partial differential equations which involve the operator. The aim of this paper is to review several already used discrete Laplace-Beltrami operators over triangulated surface and study numerically as well as theoretically their convergent behavior. We show that none of them is convergent in general, but two of them, proposed by Desbrun et al and Meyer et al, are convergent in a special case. We point out that this special case is very important in the numerical simulation of geometric partial differential equations.