您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

山东大学学报（理学版） ›› 2016, Vol. 51 ›› Issue (4): 90-98.doi: 10.6040/j.issn.1671-9352.0.2015.154

• • 上一篇    下一篇

拟线性黏弹性方程一个新的H 1-Galerkin混合有限元分析

刁群1,石东洋2   

  1. 1.平顶山学院数学与信息科学学院, 河南 平顶山 467000;2.郑州大学数学与统计学院, 河南 郑州 450001
  • 收稿日期:2015-04-14 出版日期:2016-04-20 发布日期:2016-04-08
  • 作者简介:刁群(1979— ), 女, 硕士, 讲师, 研究方向为有限元方法及其应用. E-mail:diaoqun.happy@163.com
  • 基金资助:
    国家自然科学基金资助项目(11271340)

New H 1-Galerkin mixed finite element analysis for quasi-linear viscoelasticity equation

DIAO Qun1, SHI Dong-yang2   

  1. 1. School of Mathematics and Information Science, Pingdingshan University, Pingdingshan 467000, Henan, China;
    2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, Henan, China
  • Received:2015-04-14 Online:2016-04-20 Published:2016-04-08

PDF (PC)

1061

摘要： 利用不完全双二次元Q-2和一阶BDFM元, 对拟线性黏弹性方程构造了一个新的H 1-Galerkin混合元模式。通过Bramble-Hilbert引理, 证明了单元所对应的插值算子一个新的高精度结果。 进一步地, 在半离散和一个二阶全离散格式下, 分别导出了原始变量u在H 1-模和中间变量(→overp)在 H(div)-模意义下的超逼近性质。

关键词: 拟线性黏弹性方程, Bramble-Hilbert引理, 超逼近, H 1-Galerkin混合有限元方法, 半离散和全离散格式

Abstract: A new H 1-Galerkin mixed finite element pattern for quasi-linear viscoelasticity equation is constructed using incomplete biquadratic element Q-2 and first order BDFM element. Through Bramble-Hilbert lemma, a newhigh 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 (→overp) in H(div)-norm are obtained respectively in semi-discrete and fully discrete schemes.

Key words: quasi-linear viscoelasticity equation, H 1-Galerkin mixed finite element method, superclose, semi-discrete and fully discrete schemes, Bramble-Hilbert lemma

中图分类号: 

  • O242.21
[1] LIN Qun, ZHANG Shuhua. A direct global superconvergence analysis for Sobolev and viscoelasticity type equations[J]. Appl Math, 1997, 42(1):23-34.
[2] JIN Dayong, LIU Tang, ZHANG Shuhua. Global superconvergence analysis of Wilson element for Sobolev and viscoelasticity type equations[J]. J Syst Sci Complex, 2004, 17(4):452-463.
[3] SHI Dongyang, PENG Yucheng, CHEN Shaochun. Superconvergence of a nonconforming finite element approximation to viscoelasticity type equations on anisotropic meshes[J]. Numerical Mathematics: A Journal of Chinese Universities(English Series), 2006, 15(4):375-384.
[4] 石东洋, 关宏波. 粘弹性方程的非协调变网格有限元方法[J]. 高校应用数学学报, 2008, 23(4):452-458. SHI Dongyang, GUAN Hongbo. A class of nonconforming finite element methods for viscoelasticity type equations with moving grid[J]. Appl Math J Chin Univ, 2008, 23(4):452-458.
[5] 李宏, 孙萍, 尚月强,等. 粘弹性方程全离散化有限体积元格式及数值模拟[J]. 计算数学, 2012, 34(4):413-424. LI Hong, SUN Ping, SHANG Yueqiang, et al. A fully discrete finite volume element formulation and numerical simulations for viscoelastic equations[J]. Math Numer Sin, 2012, 34(4):413-424.
[6] 李先崇, 孙萍, 安静,等.粘弹性方程一种新的分裂正定混合元法[J]. 计算数学, 2013, 35(1):49-58. LI Xianchong, SUN Ping, AN Jing, et al. A new splitting positive definite mixed finite element method for viscoelastic equation[J]. Math Numer Sin, 2013, 35(1):49-58.
[7] 彭玉成, 华沛. 粘弹性方程的一个二阶非协调有限元逼近分析[J]. 高等学校计算数学学报, 2013, 35(3):240-249. PENG Yucheng, HUA Pei. Analysis of a second order nonconforming finite element approximation to viscoelasticity type equations[J]. Numer Math J Chin Univ, 2013, 35(3):240-249.
[8] SHI Dongyang, ZHANG Buying. High accuracy analysis of the finite element method for nonlinear viscoelastic wave equations with nonlinear boundary conditions[J]. J Syst Sci Complex, 2011, 24(4):795-802.
[9] WANG Fengling, ZHAO Yanmin, SHI Dongyang. EQrot1 nonforming finite element analysis for nonlinear viscoelasticity equations[J]. Math Appl, 2013, 26(1):1-10.
[10] Pani A K. An H 1-Galerkin mixed finite element methods for parabolic partial differential equations[J]. SIAM J Numer Anal, 1998, 35(2):721-727.
[11] 郭玲, 陈焕贞. Sobolev方程的H 1-Galerkin混合有限元方法[J]. 系统科学与数学, 2006, 26(3):301-314. GUO Ling, CHEN Huanzhen. H 1-Galerkin mixed finite element method for the Sobolev equation[J]. J Sys Sci & Math Scis, 2006, 26(3):301-314.
[12] 王瑞文. 双曲型积分微分方程H 1-Galerkin混合元法的误差估计[J]. 计算数学, 2006, 28(1):19-30. WANG Ruiwen. Error estimates for H 1-Galerkin mixed finite element methods for hyperbolic type integro-differential equation[J]. Math Numer Sin, 2006, 28(1):19-30.
[13] 刘洋, 李宏, 何斯日古楞. 伪双曲型积分-微分方程H 1-Galerkin混合元法误差估计[J]. 高等学校计算数学学报, 2010, 32(1):1-20. LIU Yang, LI Hong, HE Siriguleng. Error estimates of H 1-Galerkin mixed finite element methods for pseudo-hyperbolic partial integro-differential equation[J]. Numer Math J Chin Univ, 2010, 32(1):1-20.
[14] 石东洋, 唐启立, 董晓靖. 强阻尼波动方程的H 1-Galerkin混合有限元超收敛分析[J]. 计算数学, 2012, 34(3):317-328. SHI Dongyang, TANG Qili, DONG Xiaojing. Superconvergence analysis of H 1-Galerkin mixed finite element method for strongly damped wave equations[J]. Math Numer Sin, 2012, 34(3):317-328.
[15] SHI Dongyang, LIAO Xin, TANG Qili. Highly efficient H1-Galerkin mixed finite element method(MFEM)for parabolic integro-differential equation[J]. Appl Math Mech, 2014, 35(7):897-912.
[16] 石东洋, 史艳华, 王芬玲. 四阶拋物方程H 1-Galerkin混合有限元方法的超逼近及最优误差估计[J]. 计算数学, 2014, 36(4):363-380. SHI Dongyang, SHI Yanhua, WANG Fenling. Supercloseness and the optimal order error estimates of H 1-Galerkin mixed finite element method for forth-order parabolic equation[J]. Math Numer Sin, 2014, 36(4):363-380.
[17] 陈红斌, 刘晓奇, 徐大. 粘弹性双曲型方程的H 1-Galerkin混合有限元方法[J]. 高等学校计算数学学报, 2011, 33(3):279-288. CHEN Hongbin, LIU Xiaoqi, XU Da. H 1-Galerkin mixed finite element method for the viscoelasticity wave equation[J]. Numer Math J Chin Univ, 2011, 33(3):279-288.
[18] WANG Jinfeng, LIU Yang, LI Hong. Error estimates of H 1-Galerkin mixed methods for the viscoelasticity wave equation[J]. Chin Quart J of Math, 2011, 26(1):131-137.
[19] 郝晓斌. 非协调有限元的构造及其应用[D]. 郑州:郑州大学, 2008. HAO Xiaobin. Construction and application of nonconforming finite element[D]. Zhengzhou: Zhengzhou University, 2008.
[20] 林群, 严宁宁. 高效有限元构造与分析[M]. 保定: 河北大学出版社, 1996. LIN Qun, YAN Ningning. Construction and analysis for effecitive finite element methods[M]. Baoding: Hebei University Press, 1996.
[21] Hale J K. Ordinary differential equations[M]. New York: Willey, 1969.
[1] 张厚超, 朱维钧, 王俊俊. 非线性四阶双曲方程一个低阶混合元方法的超收敛和外推[J]. 山东大学学报（理学版）, 2015, 50(12): 35-46.
[2] 樊明智, 王芬玲, 石东洋. 广义神经传播方程最低阶新混合元格式的高精度分析[J]. 山东大学学报（理学版）, 2015, 50(08): 78-89.
[3] 张亚东1,李新祥2，石东洋3. 强阻尼波动方程的非协调有限元超收敛分析[J]. 山东大学学报（理学版）, 2014, 49(05): 28-35.
[4] 孟晓然1,石东伟2. 拟线性抛物问题各向异性R-T混合元分析[J]. J4, 2012, 47(2): 36-41.
[5] 周家全,孙应德,张永胜. Burgers方程的非协调特征有限元方法[J]. J4, 2012, 47(12): 103-108.
[6] 王芬玲1,石东伟2. 非线性双曲方程Hermite型矩形元的高精度分析[J]. J4, 2012, 47(10): 89-96.
[7] 乔保民,梁洪亮. 一类非线性广义神经传播方程Adini元的超收敛分析[J]. J4, 2011, 46(8): 42-46.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发