Title: Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations,
Abstract: 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.