A hybridized weak Galerkin finite element scheme for general second-order elliptic problems

Guanrong Li; Yanping Chen; Yunqing Huang; Guanrong Li; Yanping Chen; Yunqing Huang

doi:10.3934/era.2020042

Electronic Research Archive

2020, Volume 28, Issue 2: 821-836. doi: 10.3934/era.2020042

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A hybridized weak Galerkin finite element scheme for general second-order elliptic problems

Guanrong Li ^{1
,},
Yanping Chen ^{2
,
,},
Yunqing Huang ^{3
,}

1.
School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, Guangdong, China
2.
School of Mathematical Science, South China Normal University, Guangzhou 520631, Guangdong, China
3.
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China

Received: 01 February 2020 Revised: 01 March 2020
Primary: 65N15, 65N30; Secondary: 35J50

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.
- Hybridized weak Galerkin,
- finite element methods,
- discrete gradient,
- general second-order elliptic problems,
- convergence analysis
Citation: Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems[J]. Electronic Research Archive, 2020, 28(2): 821-836. doi: 10.3934/era.2020042

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Abstract

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

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