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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2016, Vol. 31 Issue (2): 215-224    DOI:
    
A new $H^{1}$-Galerkin mixed finite element analysis for Sobolev equation
DIAO Qun1, SHI Dong-yang2, ZHANG Fang2
1. School of Math. Statis., Pingdingshan Univ., Pingdingshan 467000, China
2. School of Math. Statis., Zhengzhou Univ., Zhengzhou 450001, China
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Abstract  In this paper, $H^{1}$-Galerkin mixed finite element method for Sobolev equation is studied. A new mixed finite element pattern is constructed using incomplete biquadratic element $Q_{2}^{-}$ and first order BDFM element. Through Bramble-Hilbert lemma, high precision results of interpolation operators corresponding to unit are proved. Further, the superclose properties for the primitive variables $u$ in $H^{1}$-norm and the intermediate variable $\vec{p}$ in $H(\text{div})$-norm are obtained respectively in semi-discrete and the backward Euler fully discrete schemes.

Key words： Sobolev equation      $H^{1}$-Galerkin mixed finite element method      Bramble-Hilbert lemma      semi-discrete and fully discrete schemes      superclose     
Received: 10 April 2015      Published: 17 May 2018
CLC:  O242.21  
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DIAO Qun
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DIAO Qun, SHI Dong-yang, ZHANG Fang. A new $H^{1}$-Galerkin mixed finite element analysis for Sobolev equation. Applied Mathematics A Journal of Chinese Universities, 2016, 31(2): 215-224.

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http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2016/V31/I2/215


Sobolev方程一个新的$H^{1}$-Galerkin混合有限元分析

研究了Sobolev方程的$H^{1}$-Galerkin混合有限元方法. 利用不完全双二次元$Q_{2}^{-}$和一阶BDFM元, 建立了一个新的混合元模式, 通过Bramble-Hilbert引理, 证明了单元对应的插值算子具有的高精度结果. 进一步, 对于半离散和向后欧拉全离散格式, 分别导出了原始变量$u$在$H^{1}$-模和中间变量$\vec{p}$在$H(div)$-模意义下的超逼近性质.

关键词： Sobolev方程,  $H^{1}$-Galerkin混合有限元方法,  Bramble-Hilbert引理,  半离散和全离散格式,  超逼近 
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