稳态对流扩散方程边值问题的一种有限元求解方法

A Finite Element Method for Solving the Boundary Value Problem of the Steady Convection-Diffusion Equation

摘要: 在本文中，我们为稳态对流扩散方程边值问题设计一种有限元法。对流扩散方程边值问题与普通的边值问题不同，方程之中含有一个微小元项，它会给高阶数值方法的设计带来困难。我们首先通过设计典型的有限元法(包括线性元和二次元)来求解该边值问题，然后用MATLAB画图来比较近似解与精确解之间的实际差距，分析这两种典型的有限元法在求解该边值问题过程中所出现的问题；最后提出建议通过基于非均匀网格来改进这两种典型的有限元法，以便更好地求解这类稳态对流扩散方程边值问题。

Abstract: In this article, we aim to design a finite element method for solving the boundary value problem of the steady convection-diffusion equation. This boundary value problem is different from the general one, in which there is a small term in the equation, which will make us difficult to design a higher-order numerical method for such problem. We first design two standard finite element methods (including linear and quadratic finite element method) to solve this boundary value problem; we next use these two methods to obtain the approximated solution, and compare the approximated solution with the analytical one in Matlab; we finally propose suggestions to improve these two standard finite element methods based on nonuniform grids, in order to find a better approximation to the boundary value problem of the convection-diffusion equation.