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

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 three dimensions (3D). Following the so-called DDFV (discrete duality finite volume) approach developed by Hermeline (1998, Une méthode de volumes finis pour les équations elliptiques du second ordre. C. R. Math. Acad. Sci. Paris, 326, 1433–1436 (in French); 2000, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys., 160, 481–499) and by Domelevo & Omnès (2005, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal., 39, 1203–1249) in 3D, we consider a ‘double’ covering of a 3D domain by a rather general primal mesh and by a well-chosen ‘dual’ mesh. The associated discrete divergence operator div  is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator ∇ is defined by a 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  and ∇ 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, Andreianov et al. (2011a, On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems. HAL preprint available at http://hal.archives-ouvertes.fr/hal-00567342) to 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 partial differential equations.