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university of Provence - Aix-Marseille I |  |
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university of Provence - Aix-Marseille I | |
| HAL: hal-00346077, version 1 |
| arXiv: 0812.2097 |
| DOI: 10.1142/S0218202510004222 |
| Detailed view |  Export this paper |
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| Math. Models Methods Appl. Sci. 20, 2 (2010) 265--295 |
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| A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods |
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| Jerome Droniou 1Robert Eymard 2 |
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| (2010) |
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| We investigate the connections between several recent methods for the discretization of ani\-so\-tropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme. |
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| 1: | Institut de Mathématiques et de Modélisation de Montpellier (I3M) |
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CNRS : UMR5149 – Université Montpellier II - Sciences et techniques |
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| 2: | Laboratoire d'Etudes des Transferts d'Energie et de Matière (LETEM) |
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Université Paris-Est Marne-la-Vallée (UPEMLV) : EA2546 |
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| 3: | Laboratoire d'Analyse, Topologie, Probabilités (LATP) |
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CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III |
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| Subject | : | Mathematics/Numerical Analysis |
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| hal-00346077, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00346077 | |
| oai:hal.archives-ouvertes.fr:hal-00346077 | |
| From: Raphaele Herbin | |
| Submitted on: Thursday, 11 December 2008 09:36:16 | |
| Updated on: Friday, 29 October 2010 21:58:16 | |
