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A family of interior-penalized weak Galerkin methods for second-order elliptic equations
AIMS Mathematics ( IF 2.2 ) Pub Date : 2020-10-19 , DOI: 10.3934/math.2021030
Kaifang Liu , , Lunji Song

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.

中文翻译:

二阶椭圆方程的内部罚弱Galerkin方法族

提出并分析了内部惩罚的弱伽勒金(IPWG)有限元方法来求解二阶椭圆方程。新方法使用元素$(\ mathbb {P} _ {k},\ mathbb {P} _ {k},\ mathcal {RT} _ {k})$,尺寸为$ d = 2,3 ,建立了离散的$ H ^ 1 $ -norm和$ L ^ 2 $ -norm中的最优先验误差估计。而且,只要精确解的平滑度足够大,就可以导出$ H ^ 1 $和$ L ^ 2 $范数的超收敛性。提出了一些数值实验,以证明IPWG方法的灵活性,有效性和可靠性。在实验中,IPWG方法的收敛速率在$ L ^ 2 $范数下是最佳的,而对于多项式为偶数的情况,它们对于NIPG和IIPG而言次优。
更新日期:2020-10-19
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