Fick and Fokker–Planck problems: A finite element comparison

We consider two diffusion processes modeled by the Fick and Fokker–Planck continuity equations when the diffusion coefficient depends on the temperature and obeys an Arrhenius formula. For both problems, we present an unifying numerical framework in order to study and compare the differences and physical consequences of each approach. The numerical study is based on the theory of the finite element method, where the use of stabilization procedures becomes fundamental in order to obtain accurate solutions. In the process, we provide existence and uniqueness of solutions and present different numerical schemes for approximation. An exhaustive numerical study is performed, where optimal convergence is provided, the idea of the need of stabilization is presented, and the analysis of a problem related to real-world scenarios is detailed in depth, where we quantify the difference in both formulations and study the impact of different physical terms, through the Péclet and Damköhler numbers. All the analysis is valid under mild assumptions on data, for general two and three-dimensional domains.