On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality.

Boris Andreianov; Mostafa Bendahmane; Florence Hubert; Stella Krell

hal-00355212, version 3

On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality.

Boris Andreianov () ¹, Mostafa Bendahmane () ^2, 3, Florence Hubert () ⁴, Stella Krell () ^4, 5

(11/03/2011)

Abstract: This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in 3D. Following the approach developed by F. Hermeline and by K.~Domelevo and P. Omnès in 2D, we consider a ``double'' covering $\Tau$ of a three-dimensional domain by a rather general primal mesh and by a well-chosen ``dual'' mesh. The associated discrete divergence operator $\div^{\ptTau}$ is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator $\grad^\ptTau$ is defined by local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that $-\div^{\ptTau}$, $\grad^\ptTau$ are linked by the ``discrete duality property'', which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel [3] of this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic PDEs.

1: Laboratoire de Mathématiques (LM-Besançon)
CNRS : UMR6623 – Université de Franche-Comté
2: Centro de Investigación en Ingeniería Matemática [Concepción] (CI²MA)
Universidad de Concepción
3: Institut de Mathématiques de Bordeaux (IMB)
CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II
4: Laboratoire d'Analyse, Topologie, Probabilités (LATP)
CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
5: SIMPAF (INRIA Lille - Nord Europe)
INRIA – Université Lille I - Sciences et technologies – CNRS : UMR

hal-00355212, version 3
http://hal.archives-ouvertes.fr/hal-00355212
oai:hal.archives-ouvertes.fr:hal-00355212
From: Boris Andreianov <>
Submitted on: Friday, 11 March 2011 16:38:52
Updated on: Friday, 11 March 2011 18:27:50

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%% hal-00355212, version 3
%% http://hal.archives-ouvertes.fr/hal-00355212
@unpublished{andreianov:hal-00355212,
    hal_id = {hal-00355212},
    url = {http://hal.archives-ouvertes.fr/hal-00355212},
    title = {{On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality.}},
    author = {Andreianov, Boris and Bendahmane, Mostafa and Hubert, Florence and Krell, Stella},
    abstract = {{This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in 3D. Following the approach developed by F. Hermeline and by K.\~Domelevo and P. Omn{\`e}s in 2D, we consider a ``double'' covering $\Tau$ of a three-dimensional domain by a rather general primal mesh and by a well-chosen ``dual'' mesh. The associated discrete divergence operator $\div^{\ptTau}$ is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator $\grad^\ptTau$ is defined by local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that $-\div^{\ptTau}$, $\grad^\ptTau$ are linked by the ``discrete duality property'', which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel [3] of this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic PDEs.}},
    keywords = {Finite volume approximation; Gradient reconstruction; Discrete gradient; Discrete duality; 3D CeVe-DDFV; Consistency; Anisotropic elliptic problems; General mesh; Non-conformal mesh},
    language = {English},
    affiliation = {Laboratoire de Math{\'e}matiques - LM-Besan{\c c}on , Centro de Investigaci{\'o}n en Ingenier{\'\i}a Matem{\'a}tica [Concepci{\'o}n] - CI\up2MA , Institut de Math{\'e}matiques de Bordeaux - IMB , Laboratoire d'Analyse, Topologie, Probabilit{\'e}s - LATP , SIMPAF - INRIA Lille - Nord Europe},
    note = {This is a largely extended version with respect to version 1. Chile FONDECYT program No.7080187 },
    year = {2011},
    month = Mar,
    pdf = {http://hal.archives-ouvertes.fr/hal-00355212/PDF/AndBendHubKrell-I-preprint2011.pdf},
}

%F hal-00355212, version 3
%0 Unpublished Work
%U http://hal.archives-ouvertes.fr/hal-00355212
%2 MATH:MATH_NA;MATH:MATH_AP
%T On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality.
%A Andreianov, Boris
%A Bendahmane, Mostafa
%A Hubert, Florence
%A Krell, Stella
%+ Laboratoire de Mathématiques - LM-Besançon , Centro de Investigación en Ingeniería Matemática [Concepción] - CI²MA , Institut de Mathématiques de Bordeaux - IMB , Laboratoire d'Analyse, Topologie, Probabilités - LATP , SIMPAF - INRIA Lille - Nord Europe
%X This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in 3D. Following the approach developed by F. Hermeline and by K.~Domelevo and P. Omnès in 2D, we consider a ``double'' covering $\Tau$ of a three-dimensional domain by a rather general primal mesh and by a well-chosen ``dual'' mesh. The associated discrete divergence operator $\div^{\ptTau}$ is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator $\grad^\ptTau$ is defined by local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that $-\div^{\ptTau}$, $\grad^\ptTau$ are linked by the ``discrete duality property'', which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel [3] of this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic PDEs.
%G English
%2 2011-03-11
%K Finite volume approximation; Gradient reconstruction; Discrete gradient; Discrete duality; 3D CeVe-DDFV; Consistency; Anisotropic elliptic problems; General mesh; Non-conformal mesh
%Z MSC 65N12,65M12
%Z This is a largely extended version with respect to version 1.
%Z Chile FONDECYT program No.7080187

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