We consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions (a {\it field} and a {\it road} in a population dynamics context) and coupled through exchange terms on the road, which makes its analysis quite involved. We propose a TPFA finite volume scheme. In both the continuous and the discrete settings, we prove the exponential decay of an entropy, and thus the long time convergence to the stationary state selected by the total mass of the initial data. To deal with the problem of different dimensions, we artificially \lq\lq thicken'' the road and, then, establish a rather unconventional Poincaré-Wirtinger inequality. Numerical simulations confirm and complete the analysis, and raise new issues.