ANDREIANOV, Boris; BENDAHMANE, Mostafa; HUBERT, Florence; KRELL, Stella

doi:10.1093/imanum/drr046

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ANDREIANOV, Boris
Laboratoire de Mathématiques de Besançon (UMR 6623) [LMB]

BENDAHMANE, Mostafa
Centro de Investigación en Ingeniería Matemática [Concepción] [CI²MA]
Institut de Mathématiques de Bordeaux [IMB]

HUBERT, Florence
Laboratoire d'Analyse, Topologie, Probabilités [LATP]

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IMA Journal of Numerical Analysis. 2012-10-12, vol. 32, n° 4, p. pp.1574-1603

Oxford University Press (OUP)

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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.< Réduire

Mots clés en anglais

Non-conformal mesh

General mesh

Consistency

Anisotropic elliptic problems

Finite volume approximation

Gradient reconstruction

Discrete gradient

Discrete duality

3D CeVe-DDFV

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

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