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

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

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。

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 reconstruction type 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.

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

Key words

the nonlinear convection-diffusion equation / the characteristic method / gradient reconstruction / adaptive finite element method

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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)