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

TL;DR

This paper provides pointwise error estimates for Voronoi-based finite volume methods solving the Poisson equation on the sphere, establishing quasi-optimal convergence in the maximum norm with minimal regularity assumptions.

Contribution

It introduces pointwise error estimates for Voronoi-based finite volume methods on the sphere, linking them to finite element methods and deriving convergence rates using Green's functions.

Findings

Convergence order is quasi-optimal in maximum norm.

Error estimates are valid under minimal regularity.

Numerical results confirm theoretical predictions.

Abstract

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.