On pointwise error estimates for Voronoï-based finite volume methods for the Poisson equation on the sphere (Q6174716)

The authors provide pointwise estimates of a Voronoï-based finite volume approximation of the Laplace-Beltrami operator on Voronoï-Delaunay decompositions of the sphere. Using these estimates, a local error analysis, in the maximum norm, of the approximate solution of the Poisson equation and its gradient, is performed. The Voronoï-based finite volume method is considered as a perturbation of the finite element method. Thanks to the use of the regularized Green's functions, a quasi-optimal convergence order in the maximum-norm with minimal regularity requirements is derived and proved. Some numerical examples are presented to show numerically the convergence of the proposed method.