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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

(2011-02-20)

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

Subject

:

Mathematics/Numerical Analysis Mathematics/Analysis of PDEs

Keyword(s): Finite volume approximation – Gradient reconstruction – Discrete gradient – Discrete duality – 3D CeVe-DDFV – Consistency – Anisotropic elliptic problems – General mesh – Non-conformal mesh

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From: Boris Andreianov <>

Submitted on: Sunday, 20 February 2011 22:28:39

Updated on: Monday, 21 February 2011 18:50:41