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

Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations