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高校应用数学学报
高校应用数学学报  2017, Vol. 32 Issue (4): 437-454    
    
EFK方程一个新的低阶非协调混合有限元方法的高精度分析
张厚超1, 石东洋2
1. 平顶山学院数学与统计学院, 河南平顶山467000;
2. 郑州大学数学与统计学院, 河南郑州450001
High accuracy analysis of a new low order nonconforming mixed finite element method for the EFK equation
ZHANG Hou-chao1, SHI Dong-yang2
1. School of Mathematics and Statistics, Pingdingshan Univ., Pingdingshan 467000, China;
2. School of Mathematics and Statistics., Zhengzhou Univ., Zhengzhou 450001, China
 全文: PDF 
摘要： 对Extended Fisher-Kolmogorov(EFK)方程, 利用$EQ_{1}^{rot}$元和零阶Raviart-Thomas(R-T)元建立了一个新的非协调混合元逼近格式. 首先, 证明了半离散格式逼近解的一个先验估计并证明了逼近解的存在唯一性. 在半离散格式下, 利用上述两种元的高精度分析结果以及这个先验估计, 在不需要有限元解$u_{h}$ 属于$L^{\infty}$的条件下, 得到了原始变量$u$和中间变量$v=-\Delta u$的$H^{1}$-模以及流量$\vec{p}=\nabla u$的$(L^{2})^{2}$-模意义下$O(h^{2})$阶的超逼近性质. 同时, 借助插值后处理技术, 证明了上述变量的具有$O(h^{2})$阶的整体超收敛结果. 其次, 建立了一个新的线性化向后Euler全离散格式并证明了其逼近解的存在唯一性. 另一方面, 通过对相容误差和非线性项采取与传统误差分析不同的新的分裂技巧, 分别导出了以往文献中尚未涉及的关于$u$和$v$在$H^{1}$- 模以及$\vec{p}$在$(L^{2})^{2}$-模意义下具有$ O(h^{2}+\tau)$阶的超逼近性质, 进一步地, 借助插值后处理技术, 得到了上述变量的整体超收敛结果. 这里$h$和$\tau$分别表示空间剖分参数和时间步长. 最后, 给出了一个数值算例, 计算结果验证了理论分析的正确性.}
关键词： EFK 方程;  $EQ_{1}^{rot}$元和零阶R-T元;  半离散和全离散格式;  超逼近和超收敛    
Abstract: With the help of $EQ^{rot}_1$ element and zero order Raviart-Thomas(R-T) elements, a new nonconforming mixed finite elements approximation scheme is proposed for the extended Fisher-Kolmogorov equation(EFK). Firstly, a priori estimate of approximation solutions is proved and the existence and uniqueness are also derived for semi-discrete scheme. Based on the high accuracy analysis of two elements and the priori estimate of the finite element solutions, the superclose properties of order $O(h^{2})$ for the primitive solution $u$ and the intermediate variable $v=-\Delta u~$in~$H^{1}$-norm and flux $\vec{p}=\nabla u$ in $(L^{2})^{2}$-norm are obtained without the boundedness of numerical solution $u_{h}$ in $L^{\infty}$-norm, respectively. Meanwhile, the global superconvergent error estimates for above variables of order $O(h^{2})$ are proved through interpolated postprocessing technique. Secondly, a new linearized backward Euler full-discrete scheme is established, the existence and uniqueness of approximation scheme are proved. On the other hand, the superclose properties of order $O(h^{2}+\tau)$ for variable $u,v$ in $H^{1}$-norm and $\vec{p}$ in $(L^{2})^{2}$-norm are obtained respectively through a new splitting technique for consistent error estimate and nonlinear term, which have never been considered in the previous literature. Furthermore, the global superconvergent estimates for above variables are deduced by interpolated postprocessing technique.~Here,~$h,~\tau$~are parameters of the subdivision in space and time step, respectively. Finally, numerical results are provided to confirm the theoretical analysis.
Key words: EFK equation    $EQ_{1}^{rot}$ and zero R-T elements    Semi-discrete and full-discrete schemes    Superclose and superconvergence
收稿日期: 2016-12-07 出版日期: 2018-12-01
CLC:  O242.21  
基金资助: 国家自然科学基金(11271340; 11671369); 河南省科技厅基础与前沿项目基金(162300410082)
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引用本文:

张厚超, 石东洋. EFK方程一个新的低阶非协调混合有限元方法的高精度分析[J]. 高校应用数学学报, 2017, 32(4): 437-454.

ZHANG Hou-chao, SHI Dong-yang. High accuracy analysis of a new low order nonconforming mixed finite element method for the EFK equation. Applied Mathematics A Journal of Chinese Universities, 2017, 32(4): 437-454.

链接本文:

http://www.zjujournals.com/amjcua/CN/    或    http://www.zjujournals.com/amjcua/CN/Y2017/V32/I4/437

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