A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

doi:10.1007/s10473-020-0521-y

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1553-1562.doi: 10.1007/s10473-020-0521-y

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

祝鹏¹, 王筱沈²

1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China;
2. Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

收稿日期:2018-04-06 修回日期:2020-02-02 出版日期:2020-10-25 发布日期:2020-11-04
通讯作者: Peng ZHU E-mail:zhupeng.hnu@gmail.com
作者简介:Xiaoshen WANG,E-mail:xxwang@ualr.edu
基金资助:
The first author was supported by Zhejiang Provincial Natural Science Foundation of China (LY19A010008).

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

Peng ZHU¹, Xiaoshen WANG²

1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China;
2. Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

Received:2018-04-06 Revised:2020-02-02 Online:2020-10-25 Published:2020-11-04
Contact: Peng ZHU E-mail:zhupeng.hnu@gmail.com
Supported by:
The first author was supported by Zhejiang Provincial Natural Science Foundation of China (LY19A010008).

摘要/Abstract

摘要： This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H² equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

关键词: least square based weak Galerkin method, non-divergence form, weak Hessian operator, polygonal mesh

Abstract: This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H² equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

Key words: least square based weak Galerkin method, non-divergence form, weak Hessian operator, polygonal mesh

中图分类号:

65N15

祝鹏, 王筱沈. A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM[J]. 数学物理学报(英文版), 2020, 40(5): 1553-1562.

Peng ZHU, Xiaoshen WANG. A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1553-1562.

参考文献 11

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A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

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