Discrete Laplace-Beltrami Operator on Sphere and Optimal Spherical Triangulations

In this paper we first modify a widely used discrete Laplace Beltrami operator proposed by Meyer et al over triangular surfaces, and then establish some convergence results for the modified discrete Laplace Beltrami operator over the triangulated spheres. A sequence of spherical triangulations which is optimal in certain sense and leads to smaller truncation error of the discrete Laplace Beltrami operator and a sequence of hierarchical spherical triangulations are constructed. Truncation error bounds of the discrete Laplace Beltrami operator over the constructed triangulations are provided.