对流扩散方程一类改进的特征线修正有限元方法

１引言在地下水污染，地下渗流驱动，核污染，半导体等问题的数值模拟中，均涉及抛物型对流扩散方程（或方程组）的数值求解问题．这些对流扩散型偏微分方程（或方程组）具有共同的特点：对流的影响远大于扩散的影响，即对流占优性，对流占优性给问题的数值求解带来许多困难，因此对流占优问题的有效数值解法一直是计算数学中重要的研究内容．用通常的差分法或有限元法进行数值求解将出现数值振荡．为了克服数值振荡，提出各种迎风方法和修正的特征方法并在这些问题上得到成功的实际应用、８０年代，Ｄｏｕｇｌａｓ和Ｒｕｓｓｅｌｌ［２］等…

A KIND OF IMPROVED CHARACTERISTICS FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS

Characteristics finite element methods have been proved available for numerical approximation of convection-diffusion equations. All of the existed schemes of characteristics finite element or finite difference methods are limited to have only one order accuracy in time increment. In this paper, a new variational formulation is introduced after some important modification for convection term and diffusion term, then a class of characteristics finite element scheme is established from the variational formulation. Priori estimates are given which show that this scheme has second order accuracy in time increment. The optimal error estimates in both of H1 and L2 norm are demonstrated. This conclusion implies an essential improvement of existed schemes.