四阶抛物偏微分方程的H1-Galerkin混合元方法及数值模拟

doi:10.12286/jssx.2012.3.259

计算数学 ›› 2012, Vol. 34 ›› Issue (3): 259-274.DOI: 10.12286/jssx.2012.3.259

四阶抛物偏微分方程的H¹-Galerkin混合元方法及数值模拟

刘洋, 李宏, 何斯日古楞, 高巍, 方志朝

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).

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刘洋, 李宏, 何斯日古楞, 高巍, 方志朝. 四阶抛物偏微分方程的H¹-Galerkin混合元方法及数值模拟[J]. 计算数学, 2012, 34(3): 259-274.

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

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received:2011-08-01 Online:2012-08-15 Published:2012-08-16

到目前为止, H¹-Galerkin 混合有限元方法研究的问题仅局限于二阶发展方程. 然而对于高阶发展方程, 特别是重要的四阶发展方程问题的研究却没有出现. 本文首次提出四阶发展方程的H¹-Galerkin 混合有限元方法, 为了给出理论分析的需要, 我们考虑四阶抛物型发展方程. 通过引进三个适当的中间辅助变量, 形成四个一阶方程组成的方程组系统, 提出四阶抛物型方程的H¹-Galerkin 混合有限元方法. 得到了一维情形下的半离散和全离散格式的最优收敛阶误差估计和多维情形的半离散格式误差估计, 并采用迭代方法证明了全离散格式的稳定性. 最后, 通过数值例子验证了提出算法的可行性. 在一维情况下我们能够同时得到未知纯量函数、一阶导数、负二阶导数和负三阶导数的最优逼近解, 这一点是以往混合元方法所不能得到的.

关键词： 四阶抛物偏微分方程, H¹-Galerkin 混合元方法, 稳定性, 最优阶误差估计

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.

Key words: Fourth-order parabolic partial differential equations, H¹-Galerkin mixed element method, Stability, Optimal error estimates

MR(2010)主题分类:

65N30
65M60

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[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.

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