Title :

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

Author(s) :

Andreianov, Boris [Auteur]
Laboratoire de Mathématiques de Besançon (UMR 6623) [LMB]
Bendahmane, Mostafa [Auteur]
Centro de Investigación en Ingeniería Matemática [Concepción] [CI²MA]
Institut de Mathématiques de Bordeaux [IMB]
Hubert, Florence [Auteur]
Laboratoire d'Analyse, Topologie, Probabilités [LATP]
Krell, Stella [Auteur]
Laboratoire Jean Alexandre Dieudonné [LJAD]
Laboratoire d'Analyse, Topologie, Probabilités [LATP]
SImulations and Modeling for PArticles and Fluids [SIMPAF]

Journal title :

IMA Journal of Numerical Analysis

Pages :

pp.1574-1603

Publisher :

Oxford University Press (OUP)

Publication date :

2012-10-12

ISSN :

0272-4979

English keyword(s) :

Non-conformal mesh
General mesh
Consistency
Anisotropic elliptic problems
Finite volume approximation
Gradient reconstruction
Discrete gradient
Discrete duality
3D CeVe-DDFV

HAL domain(s) :

Mathématiques [math]/Analyse numérique [math.NA]
Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

Comment :

This is a largely extended version with respect to version 1.

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL