  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 theexponential 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 Poincar{\'e}-Wirtinger
inequality. Numerical simulations confirm and complete the analysis, and raise
new issues.
