广义对流扩散问题的稳定化有限元方法

Stabilized finite element discretizations of general convection-diffusion problems

本文介绍稳态对流扩散问题的稳定化有限元方法.该方法的主要难点在于,当对流占优时可能出现边界层,导致传统有限元方法在边界层内失去稳定性,从而产生剧烈振荡.在拟均匀网格下,稳定化有限元方法可分为两类:迎风型方法和指数拟合方法.前者利用对流速度的信息在变分形式中增加稳定化项,而后者利用边界层解的特征将指数函数引入到格式设计中.这两类方法对于设计电磁场等新型对流扩散问题的数值方法起到重要指导作用.

In this paper,we overview several stabilized finite element methods for steady-state convectiondiffusion 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 solution.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.