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.

翻译：我们考虑有界域中的所谓场地-道路扩散模型，该模型由定义在不同维度集合（在种群动力学背景下为“场地”和“道路”）上的两个抛物型偏微分方程组成，并通过道路上的交换项耦合，这使得其分析较为复杂。我们提出了一种TPFA有限体积格式。在连续和离散两种设定下，我们证明了熵的指数衰减，从而证明了初始数据总质量所选择的稳态解的长时间收敛性。为了处理不同维度的问题，我们人为地将道路“加厚”，并建立了一个相当非标准的庞加莱-维尔廷格不等式。数值模拟验证并补充了该分析，同时提出了新的问题。