A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

doi:10.1007/s42967-020-00097-z

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (3): 527-544.doi: 10.1007/s42967-020-00097-z

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A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

Ahmed Al-Taweel¹, Yinlin Dong², Saqib Hussain³, Xiaoshen Wang¹

1 Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
2 Department of Mathematics, University of Central Arkansas, Conway, AR 72035, USA;
3 Department of Mathematics and Physics, Texas A&M International University, Laredo, TX 78041, USA

收稿日期:2020-07-14 修回日期:2020-10-07 出版日期:2021-09-20 发布日期:2021-09-16
通讯作者: Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang E-mail:asaltaweel@ualr.edu;ydong5@uca.edu;saqib.hussain@tamiu.edu;xxwang@ualr.edu

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

Ahmed Al-Taweel¹, Yinlin Dong², Saqib Hussain³, Xiaoshen Wang¹

1 Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
2 Department of Mathematics, University of Central Arkansas, Conway, AR 72035, USA;
3 Department of Mathematics and Physics, Texas A&M International University, Laredo, TX 78041, USA

Received:2020-07-14 Revised:2020-10-07 Online:2021-09-20 Published:2021-09-16
Contact: Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang E-mail:asaltaweel@ualr.edu;ydong5@uca.edu;saqib.hussain@tamiu.edu;xxwang@ualr.edu

摘要/Abstract

摘要： In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the ${P_k}$-harmonic polynomial space instead of the full polynomial space ${P_k}$ is to use a much smaller number of basis functions to achieve the same accuracy when k ≥ 2. The optimal rate of convergence is derived in both H¹ and L² norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the P₂-harmonic polynomial space and using the standard P₂ polynomial space are presented.

关键词: Harmonic polynomial, Weak Galerkin finite element, Laplace equation

Abstract: In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the ${P_k}$-harmonic polynomial space instead of the full polynomial space ${P_k}$ is to use a much smaller number of basis functions to achieve the same accuracy when k ≥ 2. The optimal rate of convergence is derived in both H¹ and L² norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the P₂-harmonic polynomial space and using the standard P₂ polynomial space are presented.

Key words: Harmonic polynomial, Weak Galerkin finite element, Laplace equation

中图分类号:

Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang. A Weak Galerkin Harmonic Finite Element Method for Laplace Equation[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 527-544.

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A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

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