On pointwise error estimates for Voronoi-based finite volume methods for the Poisson equation on the sphere

In this paper, we give pointwise estimates of a Voronoi-based finite volume approximation of the Laplace-Beltrami operator on Voronoi-Delaunay decompositions of the sphere. These estimates are the basis for local error analysis, in the maximum norm, of the approximate solution of the Poisson equation and its gradient. Here, we consider the Voronoi-based finite volume method as a perturbation of the finite element method. Finally, using regularized Green's functions, we derive quasi-optimal convergence order in the maximum-norm with minimal regularity requirements. Numerical examples show that the convergence is at least as good as predicted.