AAM  >> Vol. 5 No. 1 (February 2016)

    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

  • 全文下载: PDF(567KB) HTML   XML   PP.131-142   DOI: 10.12677/AAM.2016.51018  
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作者:  

邱俊,姚世举,王汉权:云南财经大学统计与数学学院,云南 昆明

关键词:
稳态对流扩散方程边值问题有限元法非均匀网格边界层Steady Convection-Diffusion Equation Boundary Value Problem Finite Element Method Nonuniform Grids Boundary Layer

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。

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.

文章引用:
邱俊, 姚世举, 王汉权. 稳态对流扩散方程边值问题的一种有限元求解方法[J]. 应用数学进展, 2016, 5(1): 131-142. http://dx.doi.org/10.12677/AAM.2016.51018

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