对流-扩散方程的自适应变换有限元方法

Adaptive Finite Element Method for Convection -Diffusion Equation in One Dimension

讨论用于对流－扩散方程的一种自适应有限元方法、在该方法中节点位置是自适应地分布的。这种分布可理解为把原独立变量自适应地变换为新变量，从而相对于新变量，解的一阶和二阶导数一致有界。这样可得到一致收敛的结果。自适应变换通过迭代计算得到，迭代步数小于等于Ｌ，这里Ｌ满足ＮＬ－１≥Ｒｅ，Ｎ是节点个数，Ｒｅ是方程中的雷诺数。

This paper discusses a finite element method for conviction-diffusion equation in one dimension in which the grids are adaptively distributed. The method transforms adaptively the original independent variable into a new variable in order that with respect to this new variable the first and second order derivatives of the solution of the equation are uniformly bounded, and uniform convergence results could be derived. The adaptive transformation is realized by iterative procedure. It is proved that this procedure will produce the transformation in L steps, if L satisfies NL-1≥>Re, where N is the number of mesh points and Re is the Reynolds number of the equation,