A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods

Jerome Droniou, Robert Eymard, Thierry Gallouet, Raphaele Herbin

Research output: Contribution to journalArticleResearchpeer-review

130 Citations (Scopus)

Abstract

We investigate the connections between several recent methods for the discretization of anisotropic 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 methods (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. 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.
Original languageEnglish
Pages (from-to)265 - 295
Number of pages31
JournalMathematical Models and Methods in Applied Sciences
Volume20
Issue number2
DOIs
Publication statusPublished - 2010
Externally publishedYes

Cite this

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abstract = "We investigate the connections between several recent methods for the discretization of anisotropic 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 methods (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. 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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A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. / Droniou, Jerome; Eymard, Robert; Gallouet, Thierry; Herbin, Raphaele.

In: Mathematical Models and Methods in Applied Sciences, Vol. 20, No. 2, 2010, p. 265 - 295.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods

AU - Droniou, Jerome

AU - Eymard, Robert

AU - Gallouet, Thierry

AU - Herbin, Raphaele

PY - 2010

Y1 - 2010

N2 - We investigate the connections between several recent methods for the discretization of anisotropic 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 methods (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. 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.

AB - We investigate the connections between several recent methods for the discretization of anisotropic 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 methods (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. 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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DO - 10.1142/S0218202510004222

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