In this paper, we give pointwise estimates of a Vorono\"i-based finite volume approximation of the Laplace-Beltrami operator on Vorono\"i-Delaunay decompositions of the sphere. These estimates are the basis for a local error analysis, in the maximum norm, of the approximate solution of the Poisson equation and its gradient. Here, we consider the Vorono\"i-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.

翻译：本文给出了球面Voronoi-Delaunay分解上基于Voronoi的Laplace-Beltrami算子有限体积近似的逐点估计。这些估计是泊松方程及其梯度的近似解在最大范数下进行局部误差分析的基础。本文将基于Voronoi的有限体积方法视为有限元方法的扰动。最后，利用正则化格林函数，我们在最小正则性要求下推导出最大范数下的拟最优收敛阶。数值算例表明，收敛性至少与预期相符。