Optimal error bounds for the two-point flux approximation finite volume scheme
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- by Robert Eymard, Thierry Gallouët and Raphaèle Herbin;
- Math. Comp. 94 (2025), 2271-2298
- DOI: https://doi.org/10.1090/mcom/4033
- Published electronically: October 18, 2024
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Abstract:
We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to $H^1_0(\Omega )$. We define an error between the approximate solution and the exact one, involving their difference and the difference of normal gradients to the faces of the mesh. We prove that this error behaves as the sum of an interpolation error and a conformity error. As a consequence, some error bounds, depending on the regularity of the continuous solution, can be obtained for both the approximate solution and an approximate gradient post-processed from this approximate solution. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.References
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Bibliographic Information
- Robert Eymard
- Affiliation: Université Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Vallée, France
- MR Author ID: 262329
- Email: robert.eymard@univ-eiffel.fr
- Thierry Gallouët
- Affiliation: I2M UMR 7373, Aix-Marseille Université, CNRS, Ecole Centrale de Marseille, F-13453 Marseille, France
- MR Author ID: 70855
- Email: thierry.gallouet@univ-amu.fr
- Raphaèle Herbin
- Affiliation: I2M UMR 7373, Aix-Marseille Université, CNRS, Ecole Centrale de Marseille, F-13453 Marseille, France
- MR Author ID: 244425
- ORCID: 0000-0003-0937-1900
- Email: raphaele.herbin@univ-amu.fr
- Received by editor(s): April 11, 2024
- Received by editor(s) in revised form: August 15, 2024
- Published electronically: October 18, 2024
- © Copyright 2024 American Mathematical Society
- Journal: Math. Comp. 94 (2025), 2271-2298
- MSC (2020): Primary 65N30, 35K15, 47A07
- DOI: https://doi.org/10.1090/mcom/4033