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 so-called DDFV (Disctere Duality Finite Volume) approach developed by F. Hermeline and by K. Domelevo and P. Omnès in 2D, we consider a "double" covering T of a three-dimensional domain by a rather general primal mesh and by a well-chosen "dual" mesh. The associated discrete divergence operator div T is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator ∇ T 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 T , ∇ T 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.

Inf-Sup stability of the discrete duality finite volume method for the 2D Stokes problem

“…We only consider here the 2D case but it is worth noticing that DDFV schemes have been successfully extended to the 3D case in [6,7,16,1] for linear anisotropic scalar diffusion equations and in [18] for the Stokes problem.…”

“…Let size(T ) be the maximum of the diameters of the diamond cells in D. We introduce a positive number reg(T ) that measures the regularity of a given mesh and is useful to perform the convergence analysis of finite volume schemes: 1) where N and N * are the maximum of edges of each primal cell and the maximum of edges incident to any vertex. The number reg(T ) should be uniformly bounded when size(T ) → 0 for the convergence results to hold.…”

A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions

“…Since they are not our main concern here, we have chosen to neglect the issues related to the nonlinear inertia term in the Navier−Stokes equations (those difficulties can be treated as in [22] for instance), thus the system we will eventually study is the coupling between the Cahn−Hilliard equation and the unsteady Stokes problem with dynamic boundary conditions, in the context of the Boussinesq approximation for the buoyancy term. Finally, for simplicity, we concentrate on the 2D case but we emphasise that 3D versions of the DDFV schemes are available and were analysed for instance in [4,30].…”

On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

“…The present paper partially follows the guidelines of Andreianov, Boyer and Hubert [12], where the 2D DDFV method was used for discretization of nonlinear diffusion problems. But we focus on the 3D DDFV schemes from the works [3,4,14] of the authors and Karlsen and Krell. In these works, a 3D generalization of the 2D DDFV scheme, called CeVe-DDFV, was described and the discrete duality property was justified.…”