• 论文 •

四阶抛物偏微分方程的H1-Galerkin混合元方法及数值模拟

1. 内蒙古大学数学科学学院, 呼和浩特 010021
• 收稿日期:2011-08-01 出版日期:2012-08-15 发布日期:2012-08-16
• 基金资助:

国家自然科学基金(11061021),内蒙古自治区高等学校科学研究基金(NJZZ12011; NJ10006; NJ10016),内蒙古大学高层次人才引进科研项目(125119；Z200901004),内蒙古大学青年科学基金(ND0702).

Liu Yang, Li Hong, He Siriguleng, Gao Wei, Fang Zhichao. H1-GALERKIN MIXED ELEMENT METHOD AND NUMERICAL SIMULATION FOR THE FOURTH-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2012, 34(3): 259-274.

H1-GALERKIN MIXED ELEMENT METHOD AND NUMERICAL SIMULATION FOR THE FOURTH-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Liu Yang, Li Hong, He Siriguleng, Gao Wei, Fang Zhichao

1. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
• Received:2011-08-01 Online:2012-08-15 Published:2012-08-16

So far, the H1-Galerkin mixed finite element method was applied to many secondorder evolution equations. However, the H1-Galerkin mixed method for the higher-order evolution equations, especially, for fourth-order evolution equations have not been studied in the literature. In this paper, we first proposed the H1-Galerkin mixed method for fourthorder evolution equation. For the need of the analysis of theories, we consider the fourthorder parabolic evolution equation. By introducing three auxiliary variables, the first-order system of four equations is formulated, and the H1-Galerkin mixed finite element method for fourth-order parabolic equation is proposed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates are derived for semidiscrete scheme for several space dimensions, and the stability for fully discrete scheme is proved by the iteration method. Finally, some numerical results are provided to illustrate the effectiveness of our method. Optimal approximate solutions for the scalar unknown, first derivative, negative second derivative and negative third derivative are obtained, which can’t be derived by the other mixed methods.

MR(2010)主题分类:

()
 [1] Pao C V. Numerical methods for fourth-order nonlinear elliptic boundary value problems[J]. Numer.Methods Partial Differential Equations, 2001, 17: 347-368.[2] Chen Z. Analysis of expanded mixed methods for fourth-order elliptic problems[J]. Numer. MethodsPartial Differential Equations, 1997, 13: 483-503.[3] Li J C. Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions[J]. Adv. Comput. Math., 2005, 23: 21-30.[4] Li J C. Optimal convergence analysis of mixed finite element methods for fourth-order elliptic andparabolic problems[J]. Numer Methods Partial Differential Equations, 2006, 22: 884-896.[5] 李宏, 刘洋. 一类四阶抛物型积分-微分方程的混合间断时空有限元方法[J]. 计算数学, 2007, 29(4): 413-420.[6] 罗振东. 混合有限元方法基础及其应用[M]. 北京: 科学出版社, 2006.[7] 张铁. Cahn-Hilliard方程的有限元分析[J]. 计算数学, 2006, 28(3): 281-292.[8] 安荣, 李开泰. 四阶障碍问题的稳定化混合有限元方法[J]. 应用数学学报,2009, 32(6): 1068-1078.[9] 石东洋, 彭玉成. 四阶特征值问题的各向异性有限元方法[J]. 工程数学学报, 2008, 25(6): 1029-1034.[10] Pani A K. An H1-Galerkin mixed finite element method for parabolic partial differential equations[J]. SIAM J. Numer. Anal., 1998, 35: 712-727.[11] Pani A K, Fairweather G. H1-Galerkin mixed finite element methods for parabolic partial integrodifferentialequations[J]. IMA Journal of Numerical Analysis, 2002, 22: 231-252.[12] Shi D Y, Wang H H. An H1-Galerkin nonconforming mixed finite element method for integrodifferentialequation of parabolic type[J]. Journal of Mathematical Research and Exposition. 2009,29(5): 871-881.[13] 王瑞文. 双曲型积分微分方程的H1-Galerkin混合有限元方法误差估计[J]. 计算数学, 2006, 28(1): 19-30.[14] Pani A K, Sinha R K, Otta A K. An H1-Galerkin mixed method for second order hyperbolicequations[J]. Inter. J. Numer. Anal. Model., 2004, 1(2): 111-129.[15] Liu Y, Li H. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl.Math. Comput., 2009, 212(2): 446-457.[16] Liu Y, Li H, Wang J F. Error estimates of H1-Galerkin mixed finite element method forSchrödinger equation[J]. Appl. Math. J. Chinese Univ., 2009, 24(1): 83-89.[17] 郭玲, 陈焕贞. Sobolev 方程的H1-Galerkin混合有限元方法[J]. 系统科学与数学, 2006, 26(3): 301-314.[18] Guo Ling, Chen Huanzhen. H1-Galerkin mixed finite element method for the regularized longwave equation[J]. Computing, 2006, 77: 205-221.[19] Pany A K, Nataraj N, Singh S. A new mixed finite element method for Burgers’ equation[J]. J.Appl. Math. Computing, 2007, 23(1-2): 43-55.[20] 刘洋, 李宏. 四阶强阻尼波方程的新混合元方法[J]. 计算数学, 2010, 32(2): 157-170.[21] Wheeler M F. A priori L2-error estimates for Galerkin approximations to parabolic differentialequations[J]. SIAM J. Numer. Anal., 1973, 10: 723-749.
 [1] 余妍妍, 代新杰, 肖爱国. 非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性[J]. 计算数学, 2022, 44(1): 19-33. [2] 高兴华, 李宏, 刘洋. 分布阶扩散—波动方程的有限元解的误差估计[J]. 计算数学, 2021, 43(4): 493-505. [3] 包学忠, 胡琳. 随机变延迟微分方程平衡方法的均方收敛性与稳定性[J]. 计算数学, 2021, 43(3): 301-321. [4] 邱泽山, 曹学年. 带漂移的单侧正规化回火分数阶扩散方程的Crank-Nicolson拟紧格式[J]. 计算数学, 2021, 43(2): 210-226. [5] 尚在久, 宋丽娜. 关于辛算法稳定性的若干注记[J]. 计算数学, 2020, 42(4): 405-418. [6] 洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309. [7] 胡冬冬, 曹学年, 蒋慧灵. 带非线性源项的双侧空间分数阶扩散方程的隐式中点方法[J]. 计算数学, 2019, 41(3): 295-307. [8] 盛秀兰, 赵润苗, 吴宏伟. 二维线性双曲型方程Neumann边值问题的紧交替方向隐格式[J]. 计算数学, 2019, 41(3): 266-294. [9] 杨晋平, 李志强, 闫玉斌. 求解Riesz空间分数阶扩散方程的一种新的数值方法[J]. 计算数学, 2019, 41(2): 170-190. [10] 王志强, 文立平, 朱珍民. 时间延迟扩散-波动分数阶微分方程有限差分方法[J]. 计算数学, 2019, 41(1): 82-90. [11] 丛玉豪, 胡洋, 王艳沛. 含分布时滞的时滞微分系统多步龙格-库塔方法的时滞相关稳定性[J]. 计算数学, 2019, 41(1): 104-112. [12] 张根根, 唐蕾, 肖爱国. 求解刚性Volterra延迟积分微分方程的隐显单支方法的稳定性与误差分析[J]. 计算数学, 2018, 40(1): 33-48. [13] 陈丰, 吴峻峰. 分布式通信响应优化问题及其内点法求解[J]. 计算数学, 2017, 39(4): 378-392. [14] 肖飞雁, 李旭旭, 陈飞盛. 非线性延迟积分微分方程连续Runge-Kutta方法的稳定性分析[J]. 计算数学, 2017, 39(1): 1-13. [15] 王涛, 刘铁钢. 求解对流扩散方程的一致四阶紧致格式[J]. 计算数学, 2016, 38(4): 391-404.