  In this contribution we analyze the large time behavior of a family of
nonlinear finite volume schemes for anisotropic convection-diffusion equations
set in a bounded bidimensional domain and endowed with either Dirichlet and /
or no-flux boundary conditions. We show that solutions to the two-point flux
approximation (TPFA) and discrete duality finite volume (DDFV) schemes under
consideration converge exponentially fast toward their steady state. The
analysis relies on discrete entropy estimates and discrete functional
inequalities. As a biproduct of our analysis, we establish new discrete
Poincar\'e-Wirtinger, Beckner and logarithmic Sobolev inequalities. Our
theoretical results are illustrated by numerical simulations.
