A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

doi:10.1007/s42967-020-00071-9

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (1): 91-105.doi: 10.1007/s42967-020-00071-9

• ORIGINAL PAPER • 上一篇

A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

Ming Cui¹, Xiu Ye², Shangyou Zhang³

1 College of Applied Science, Beijing University of Technology, Beijing, China;
2 Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
3 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

收稿日期:2019-10-30 修回日期:2020-04-09 发布日期:2021-03-15
通讯作者: Shangyou Zhang, szhang@udel.edu;Ming Cui, mingcui@bjut.edu.cn;Xiu Ye, xxye@ualr.edu E-mail:szhang@udel.edu;mingcui@bjut.edu.cn;xxye@ualr.edu

A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

Ming Cui¹, Xiu Ye², Shangyou Zhang³

1 College of Applied Science, Beijing University of Technology, Beijing, China;
2 Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
3 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Received:2019-10-30 Revised:2020-04-09 Published:2021-03-15
Contact: Shangyou Zhang, szhang@udel.edu;Ming Cui, mingcui@bjut.edu.cn;Xiu Ye, xxye@ualr.edu E-mail:szhang@udel.edu;mingcui@bjut.edu.cn;xxye@ualr.edu

摘要/Abstract

摘要： A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete H² norm are established for the corresponding finite element solutions. Error estimates in the L² norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.

关键词: Finite element methods, Weak Laplacian, Biharmonic equations, Polytopal meshes

Abstract: A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete H² norm are established for the corresponding finite element solutions. Error estimates in the L² norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.

Key words: Finite element methods, Weak Laplacian, Biharmonic equations, Polytopal meshes

中图分类号:

Ming Cui, Xiu Ye, Shangyou Zhang. A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes[J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 91-105.

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A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

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