JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (12): 35-46.doi: 10.6040/j.issn.1671-9352.0.2014.424

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Superconvergence and extrapolation of a lower order mixed finite method for nonlinear fourth-order hyperbolic equation

ZHANG Hou-chao, ZHU Wei-jun, WANG Jun-jun   

  1. School of Mathematics and Informatics, Pingdingshan University, Pingdingshan 467000, Henan, China
  • Received:2014-09-22 Revised:2015-03-06 Online:2015-12-20 Published:2015-12-23

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Abstract: With the help of the bilinear element Q11 and the Q01×Q10 element, a lower order conforming mixed finite element approximation scheme is proposed for nonlinear fourth-order hyperbolic equation. Firstly, the existence and uniqueness of approximation solution are proved. Secondly, Based on the known high accuracy results of the about two elements, by use of derivative delivery techniques, the superclose with order O(h2) for both scalar unknown u and the diffusion term v=-Δu in H1-norm and the flux p=-∇u in L2-norm are derived, respectively. Moreover, the global superconvergence is obtained through interpolation post-processing technique. Finally, throught constructing a new asymptotic expansion formula of Q01×Q10 element and a suitable extrapolation scheme, the extrapolation solutions with order O(h3) are derived. Here, h is the subdivision parameter for the space.

Key words: superclose, extrapolation, superconvergence, nonlinear fourth-order hyperbolic equation, mixed finite element methods

CLC Number: 

  • O242.21
[1] LIN Qun, WU Yonghong, LAI Shaoyong. On global solution of an initial boundary value problem for a class of damped nonlinear equation[J]. Nonlinear Analysis, 2008, 69(12):4340-4351.
[2] 徐润章, 刘博为. 四阶强阻尼非线性波动方程解的整体存在性与不存在性[J]. 数学年刊, 2011, 32A(3):267-276. XU Runzhang, LIU Bowei. Global existence and nonexistence of solution for fourth order strongly damped nonlinear wave equations[J]. Chinese Annals of Mathematics, 2011, 32A(3):267-276.
[3] YANG Zhijian. Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term[J]. J Differential Equations, 2003, 187(2):520-540.
[4] AYTEKIN G. On the numerical solution of quasilinear wave equation with strong dissipative term[J]. Appl Math Mech: English Ed, 2004, 25(7):806-811.
[5] 王震, 张立伟. 一类四阶波动方程的有限差分方法[J]. 山东科技大学学报:自然科学版, 2007, 26(4):88-91. WANG Zhen, ZHANG Liwei. Finite differential approximation for a class of fourth order wave equation[J]. Journal of Shandong University of Science: Natural Science, 2007, 26(4):88-91.
[6] 张亚东, 李新祥, 石东洋. 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(5):28-35. ZHANG Yadong, LI Xinxiang, SHI Dongyang. Superconvergence analysis of a nonconforming finite element for strongly damped wave equations[J]. Journal of Shandong University:Natural Science, 2014, 49(5):28-35.
[7] 陈金环, 王黎娜, 石东洋. 非线性四阶双曲方程的非协调有限元分析[J]. 河南师范大学学报:自然科学版, 2012, 40(1):1-6. CHEN Jinhuan, WANG Lina, SHI Dongyang. A nonconforming element for nonlinear fourth order hyperbolic equation[J]. Journal of Henan Normal University:Natural Science, 2012, 40(1):1-6.
[8] 刘洋, 李宏. 四阶强阻尼波动方程的新混合元法[J]. 计算数学, 2010, 32(2):157-170. LIU Yang, LI Hong. A new mixed finite element method for fourth order heavy damping wave equation[J]. Mathematica Numerica Sinica, 2010, 32(2):157-170.
[9] 石东洋, 唐启立, 董晓静. 强阻尼波动方程的H1-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 H1-Galerkin mixed finite element method for strongly damped wave equations[J]. Mathematica Numerica Sinica, 2012, 34(3):317-328.
[10] 陈绍春, 陈红如. 二阶椭圆问题新的混合元格式[J]. 计算数学, 2010, 32(2):213-218. CHEN Shaochun, CHEN Hongru. New mixed element schemes for second order elliptic problem[J]. Mathematica Numerica Sinica, 2010, 32(2):213-218.
[11] 史峰, 于佳平, 李开泰. 椭圆方程的一种新型混合有限元格式[J]. 工程数学学报, 2011, 28(2):231-237. SHI Feng, YU Jiaping, LI Kaitai. A new mixed finite element scheme for elliptic equations[J]. Chinese Journal of Engineering Mathematics, 2011, 28(2):231-237.
[12] 石东洋, 李明浩. 二阶椭圆问题一种新格式的高精度分析[J]. 应用数学学报, 2014, 37(1):45-58. SHI Dongyang, LI Minghao. High accuracy analysis of new schemes for second order elliptic problem for recurrent event data[J]. Acta Mathematicae Applicatae Sinica, 2014, 37(1):45-58.
[13] 石东洋, 张亚东. 抛物型方程一个新的非协调混合元超收敛分析与外推[J]. 计算数学, 2013, 35(4):337-352. SHI Dongyang, ZHANG Yadong. Superconvergence and extrapolation analysis of a new nonconforming mixed finite element approximation for parabolic equation[J]. Mathematica Numerica Sinica, 2013, 35(4):337-352.
[14] SHI Dongyang, ZHANG Yadong. High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations[J]. Appl Math Comput, 2011, 218(7):3176-3186.
[15] SHI Dongyang, LI Minghao. Superconvergence analysis of a stable conforming rectangular mixed finite elements for the linear elasticity problem[J]. J Comput Math, 2014, 32(2):205-214.
[16] LIU Yang, FANG Zhichao, LI Hong, et al. A coupling method based on new MFE and FE for fourth-order parabolic equation[J]. J Appl Math Comput, 2013, 43(1):249-269.
[17] 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, 1996:1-13. LIN Qun, YAN Ningning. Efficient finite element analysis and structure[M]. Baoding: Hebei University Press, 1996.
[18] HALE J K. Ordinary differential equations[M]. New York: Willey-Interscience, 1969:12-30.
[19] 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, 24:795-802.
[20] 史艳华, 石东洋. Sobolev方程新混合元方法的高精度分析[J]. 系统科学与数学, 2014, 34(4):452-463. SHI Yanhua, SHI Dongyang. High accuracy analysis of a new mixed finite element method for Sobolev equation[J]. J Sys Sci and Math Scis, 2014, 34(4):452-463.
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