Stabilized finite element discretizations of general convection-diffusion problems

In this paper, we overview several stabilized finite element methods for steady-state convection-diffusion problems. The main challenge lies in the occurrence of boundary layers when convection dominates, which leads to the loss of stability of traditional finite element methods within the boundary layers, resulting in severe oscillations. Under a quasi-uniform grid, stabilized finite element methods can be classified into two categories: upwind methods and exponential fitting methods. The former incorporates stabilization terms into the variational form based on the convection information, while the latter introduces exponential functions into the scheme based on the characteristics of the boundary layer 解. These two types of methods play an important guiding role in the design of the numerical schemes for new convection-diffusion problems, such as electromagnetic convection-diffusion problems.