基于有限元方法求解偏微分方程数值解的教学方法研究  

Research on Teaching Methods for Solving Numerical Solutions of Partial Differential Equations Based on Finite Element Method

作　　者：孟宜成 吴正飞[1] MENG Yi-cheng;WU Zheng-fei(Finance and Mathematics School,Huainan Normal University,Huainan,Anhui 232001)

机构地区：[1]淮南师范学院数学与金融学院,安徽淮南232001

出　　处：《集宁师范学院学报》2023年第3期22-27,共6页Journal of Jining Normal University

基　　金：安徽淮南师范学院科学研究一般项目“基于Lagrange坐标下CHO高阶交通流数学模型的研究”(2020XJYB018)。

摘　　要：本文以椭圆方程为例,通过有限元方法,对其求解偏微分方程数值解的教学方法进行了研究,对求解椭圆型偏微分方程特征值问题的弱有限元方法进行了总结。该方法可将特征值渐近下界估计给出,无需对特殊有限元空间进行构造,无需误差估计进行后验。弱有限元方法给出特征值渐近下界,对基于插值后处理获得的特征值上界方法进行,利用该算法在只求解一次特征值问题条件下可同时获得特征值上界估计、下界估计。在求解偏微分方程特征值问题时,弱有限元方法可获得高阶下界估计的性质。此类方法可对椭圆型偏微分方程特征值等问题进行应用,同时,使用Matlab工具进行有限元方法的计算,从而可获得较好的教学效果。Taking elliptic equation as an example,this paper studies the teaching method of solving numerical solution of partial differential equation by finite element method in teaching,and summarizes the weak finite element method for solving eigenvalue problem of elliptic partial differential equation.This method can give the asymptotic lower bound estimation of eigenvalue without constructing special finite element space and posterior error estimation.The asymptotic lower bound of eigenvalues is given by the weak finite element method,and the upper bound method of eigenvalues based on interpolation post-processing is carried out.By using this algorithm,the upper bound estimation and lower bound estimation of eigenvalues can be obtained simultaneously under the condition of solving the eigenvalue problem only once.When solving the eigenvalue problem of partial differential equations,the weak finite element method can obtain the properties of higher-order lower bound estimation.This kind of method can be applied to the eigenvalue of elliptic partial differential equation,and at the same time,the finite element method is calculated by Matlab tool,so that better teaching effect can be obtained.

关 键 词：教学方法 数值解 偏微分方程 有限元方法 Matlab工具 

分 类 号：O175.25[理学—数学]