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《山东大学学报(理学版)》 ›› 2026, Vol. 61 ›› Issue (2): 1-9.doi: 10.6040/j.issn.1671-9352.0.2024.109

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二维非线性对流扩散方程基于梯度重构的自适应有限元方法

苗菁菁,孙同军*   

  1. 山东大学数学学院, 山东 济南 250100
  • 发布日期:2026-02-13
  • 通讯作者: 孙同军(1970— ),男,教授,博士生导师,博士,研究方向为偏微分方程最优控制问题的数值解法. E-mail:tjsun@sdu.edu.cn
  • 作者简介:苗菁菁(2001— ),女,硕士研究生,研究方向为偏微分方程最优控制问题的数值解法. E-mail:202311824@mail.sdu.edu.cn*通信作者:孙同军(1970— ),男,教授,博士生导师,博士,研究方向为偏微分方程最优控制问题的数值解法. E-mail:tjsun@sdu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(11871312);山东省自然科学基金资助项目(ZR2023MA086)

Adaptive finite element method for two-dimensional nonlinear convection-diffusion equations based on gradient reconstruction

MIAO Jingjing, SUN Tongjun*   

  1. School of Mathematics, Shandong University, Jinan 250100, Shandong, China
  • Published:2026-02-13

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摘要： 对一类二维非线性对流扩散方程提出一种自适应有限元方法。采用特征线法处理方程对流项,有效解决因对流占优性而产生的数值震荡和数值弥散等问题。设计了一种基于梯度重构型后验误差估计的自适应有限元算法,在标准有限元方法的基础上进一步调整网格、提高精度。最后进行数值实验,验证方法的有效性。

关键词: 非线性对流扩散方程, 特征线法, 梯度重构, 自适应有限元方法

Abstract: An adaptive finite element method is proposed for a class of two-dimensional nonlinear convection-diffusion equations. The characteristic method is used to deal with the convection term of the equation, and the numerical oscillation and numerical dispersion caused by the convection dominant are effectively solved. An adaptive finite element algorithm based on a posterior error estimates of gradient reconstructiontype is designed to further adjust the mesh and improve the precision on the basis of standard finite element method. Finally, numerical experiments are carried out to verify the effectiveness of the proposed method.

Key words: the nonlinear convection-diffusion equation, the characteristic method, gradient reconstruction, adaptive finite element method

中图分类号: 

  • O241
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