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

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。

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.

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

参考文献

[1] Atkinson, K. and Han, W. (2001) Theoretical Numerical Analysis. Springer, New York.

[2] 陆金甫, 关治. 偏微分数值解法[M]. 第二版. 北京: 清华大学出版社, 2008.

[3] Bernardi, C., Maday, Y. and Rapetti, F. (2004) Discréti-sations variationnelles de problèmes aux limites elliptiques. Collection S.M.A.I. Mathematiques et, Applications. Springer, Paris, Vol. 45.

[4] 李人宪. 有限元法基础[M]. 第二版. 北京: 国防工业出版社, 2004.

[5] 毕超. 计算流体力学有限元方法及其编程详解[M]. 北京: 机械工业出版社, 2013.

[6] Brezzi, F. and Russo, A. (1994) Choosing Bubbles for Advection-Diffusion Problems. Mathematical Models and Methods in Applied Sciences, 4, 571.
http://dx.doi.org/10.1142/S0218202594000327

[7] 王焕定, 陈少峰, 边文凤. 有限单元法基础及MATLAB编程[M]. 北京: 高等教育出版社, 2012.

[8] Lucquin, B. and Pironneau, O. (1998) Introduction to Scientific Computing. Willey, Chichester.

分享
Top