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In this paper, we give pointwise estimates of a Vorono\"i-based finite volumeapproximation of the Laplace-Beltrami operator on Vorono\"i-Delaunaydecompositions of the sphere. These estimates are the basis for a local erroranalysis, in the maximum norm, of the approximate solution of the Poissonequation and its gradient. Here, we consider the Vorono\"i-based finite volumemethod as a perturbation of the finite element method. Finally, usingregularized Green's functions, we derive quasi-optimal convergence order in themaximum-norm with minimal regularity requirements. Numerical examples show thatthe convergence is at least as good as predicted.

On pointwise error estimates for Vorono\"i-based finite volume methods for the Poisson equation on the sphere