[1]魏 涛,安 静*.4阶方程基于混合格式的一种有效的Legendre-Galerkin逼近[J].江西师范大学学报(自然科学版),2023,(06):551-557+570.[doi:10.16357/j.cnki.issn1000-5862.2023.06.01]
　WEI Tao,AN Jing*.The Efficient Legendre-Galerkin Approximation Based on Mixed Schemes for Fourth-Order Equations[J].Journal of Jiangxi Normal University:Natural Science Edition,2023,(06):551-557+570.[doi:10.16357/j.cnki.issn1000-5862.2023.06.01]
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4阶方程基于混合格式的一种有效的Legendre-Galerkin逼近()
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《江西师范大学学报》（自然科学版）[ISSN:1006-6977/CN:61-1281/TN]

卷:
期数:
2023年06期
页码:
551-557+570
栏目:
数学与应用数学
出版日期:
2023-11-25

文章信息/Info

Title:
The Efficient Legendre-Galerkin Approximation Based on Mixed Schemes for Fourth-Order Equations
文章编号:
1000-5862(2023)06-0551-07
作者:
魏 涛; 安 静*
(贵州师范大学数学科学学院,贵州 贵阳 550025)
Author(s):
WEI Tao; AN Jing*
(School of Mathematical Sciences,Guizhou Normal University,Guiyang Guizhou 550025,China)
关键词:
4阶问题;  混合格式;  Legendre-Galerkin逼近;  误差估计
Keywords:
fourth order problem;  mixed scheme;  Legendre-Galerkin approximation;  error estimation
分类号:
O 241.8
DOI:
10.16357/j.cnki.issn1000-5862.2023.06.01
文献标志码:
A
摘要:
该文提出了4阶方程基于混合格式的一种有效的Legendre-Galerkin 逼近.首先,通过引入一个辅助函数,将原问题化为等价的2阶混合格式.再引入一类适当的Sobolev空间及其逼近空间,建立了2阶混合格式的弱形式和相应的离散格式; 其次,利用在非一致带权Sobolev空间中正交投影算子的逼近性质,证明了逼近解的误差估计; 另外,利用Legendre多项式的正交性质,构造了在逼近空间中的一组适当的基函数,使得在离散混合变分形式中的质量矩阵和刚度矩阵都是稀疏的,从而能够用共轭梯度法快速地求解.最后,给出了一些数值算例, 数值结果表明了所提出的算法的有效性和高精度性.
Abstract:
In this paper,the efficient Legendre-Galerkin approximation based on a mixed scheme is proposed for fourth-order equations.Firstly,by introducing an auxiliary function,the original problem is transformed into an equivalent second-order mixed scheme.Secondly,by introducing a class of suitable Sobolev spaces and their approximation spaces,the weak forms of the second-order mixed scheme and the corresponding discrete scheme are established.Then by using the approximation properties of orthogonal projection operators in non-uniform weighted Sobolev spaces,the error estimates of the approximation solutions is proved.In addition,the orthogonality of Legendre polynomials is used to construct a set of appropriate basis functions of the approximation space,so that the mass matrix and stiffness matrix in the discrete mixed variational form are all sparse,which can be quickly solved by using the conjugate gradient method.Finally,some numerical examples are given,and the numerical results show the effectiveness and high precision of our algorithm.

参考文献/References:

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备注/Memo

备注/Memo:
收稿日期:2023-09-18
基金项目:国家自然科学基金(11661022),贵州省科技计划课题(黔科合平台人才〔2017〕5726-39)和贵州师范大学学术新苗基金(黔师新苗〔2021〕A04 号)资助项目.
更新日期/Last Update: 2023-11-25