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

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.
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IMA J. Numer. Analysis, 2012, 32 (4), pp.1574-1603. 〈10.1093/imanum/drr046〉
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Boris Andreianov, Mostafa Bendahmane, Florence Hubert, Stella Krell. On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality.. IMA J. Numer. Analysis, 2012, 32 (4), pp.1574-1603. 〈10.1093/imanum/drr046〉. 〈hal-00355212v3〉

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