We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

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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.