Section: New Results

Design and analysis of advanced finite volumes schemes

The fact that a numerical method is able to handle nonlinear test functions in its numerical analysis is crucial in order to ensure its physical relevance, and consequently its good behavior.

In [15] , C. Cancès and C. Guichard proposed a first nonlinear numerical method to solve possibly degenerate parabolic equations with anisotropy on general simplicial meshes. The nonlinear control volume finite element (CVFE) scheme is based on P1 finite elements with mass-lumping combined with a tricky upwinding of the mobilities. The method has the remarkable property of preserving the positivity of the solutions. Moreover, it ensures the decay of the physical entropy. Its convergence is proved in [15] and numerical results are exhibited. In particular, they show that the method is first order accurate in space in standard situations, but can lack robustness w.r.t. the anisotropy in some particularly unfavorable situations.

This drawback was corrected by C. Cancès and C. Guichard in [35] , where a second order in space method based on the so-called VAG scheme  [57] was proposed. This method is able to handle very general grids, heterogeneous data and strong anisotropy ratios. Moreover, it preserves at the discrete level the variational structure of the continuous problem, yielding the nonlinear stability of the scheme. A complete convergence analysis was performed in [35] . The numerical results presented in [35] show that the robustness default of the first nonlinear method [15] has been corrected.

In [36] , C. Cancès et al. proposed and analyzed a nonlinear CVFE scheme for a degenerate Keller-Segel model with anisotropic and heterogeneous diffusion tensors. The scheme is based on the one proposed in [15] . The convergence of the scheme is proved under very general assumptions. Finally, some numerical experiments are carried out to prove the ability of the scheme to tackle degenerate anisotropic and heterogeneous diffusion problems over general meshes without jeopardizing the positivity of the solutions.

In [17] , C. Chainais-Hillairet, A. Jüngel and S. Schuchnigg prove the time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions in the entropy sense. The algebraic or exponential decay rates are computed explicitly. In particular, the numerical scheme dissipates all zeroth-order entropies which are dissipated by the continuous equation. The proofs are based on novel continuous and discrete generalized Beckner inequalities.

In [18] , C. Chainais-Hillairet, A. Jüngel and P. Shpartko propose and analyze a numerical scheme for a spinorial matrix-diffusion model for semiconductors. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin-vector densities, coupled to the Poisson equation for the electric potential. The main features of the numerical scheme are the preservation of nonnegativity and $L^{\infty }$ bounds of the densities and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. The fundamental ideas are reformulations using spin-up and spin-down densities and certain projections of the spin-vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic-layer field-effect transistor in two space dimensions are presented.

In [32] , M. Bessemoulin-Chatard and C. Chainais-Hillairet study the large–time behavior of a numerical scheme discretizing drift–diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme which allows to consider both linear or nonlinear pressure laws. They study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform-in-time $L^{\infty }$ estimates for numerical solutions, which are then discussed.