A family of interior-penalized weak Galerkin methods for second-order elliptic equations

Kaifang Liu; Lunji Song; Kaifang Liu; Lunji Song

doi:10.3934/math.2021030

AIMS Mathematics

2021, Volume 6, Issue 1: 500-517. doi: 10.3934/math.2021030

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Research article

A family of interior-penalized weak Galerkin methods for second-order elliptic equations

Kaifang Liu ,
Lunji Song ^,

School of Mathematics and Statistics and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China

Received: 28 August 2020 Accepted: 12 October 2020 Published: 19 October 2020
MSC : 65N15, 65N30, 35J15

Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. The new methods employ the element $(\mathbb{P}_{k}, \mathbb{P}_{k}, \mathcal{RT}_{k})$, with dimensions of space $d = 2, 3$, and the optimal a priori error estimates in discrete $H^1$-norm and $L^2$-norm are established. Moreover, provided enough smoothness of the exact solution, superconvergence in $H^1$ and $L^2$ norms can be derived. Some numerical experiments are presented to demonstrate flexibility, effectiveness and reliability of the IPWG methods. In the experiments, the convergence rates of the IPWG methods are optimal in $L^2$-norm, while they are suboptimal for NIPG and IIPG if the polynomial degree is even.
- interior-penalized weak Galerkin,
- finite element methods,
- second-order elliptic equation,
- superconvergence
Citation: Kaifang Liu, Lunji Song. A family of interior-penalized weak Galerkin methods for second-order elliptic equations[J]. AIMS Mathematics, 2021, 6(1): 500-517. doi: 10.3934/math.2021030

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Abstract

Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. The new methods employ the element $(\mathbb{P}_{k}, \mathbb{P}_{k}, \mathcal{RT}_{k})$, with dimensions of space $d = 2, 3$, and the optimal a priori error estimates in discrete $H^1$-norm and $L^2$-norm are established. Moreover, provided enough smoothness of the exact solution, superconvergence in $H^1$ and $L^2$ norms can be derived. Some numerical experiments are presented to demonstrate flexibility, effectiveness and reliability of the IPWG methods. In the experiments, the convergence rates of the IPWG methods are optimal in $L^2$-norm, while they are suboptimal for NIPG and IIPG if the polynomial degree is even.

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