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

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In this paper, we give pointwise estimates of a Voronoï-based finite volume approximation of the Laplace-Beltrami operator on Voronoï-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 Voronoï-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.

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Acknowledgements

This work is in memory of Saulo R. M. Barros, who participated in this work but tragically passed away in July 2021, prior to its conclusion. The work presented here was supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) through grant 2016/18445-7 and 2021/06176-0 and also by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES), Finance Code 001. Finally, we thank the anonymous referees for carefully reading the manuscript and for perceptive comments, which have led to an improvement in the quality of the paper.

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  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR

    Leonardo A. Poveda

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil

    Pedro Peixoto

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  1. Leonardo A. Poveda
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Correspondence to Leonardo A. Poveda.

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Poveda, L.A., Peixoto, P. On pointwise error estimates for Voronoï-based finite volume methods for the Poisson equation on the sphere. Adv Comput Math 49, 36 (2023). https://doi.org/10.1007/s10444-023-10041-3

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