非线性色散耗散波动方程双线性元的高精度分析

数学物理学报 ›› 2014, Vol. 34 ›› Issue (6): 1599-1610.

非线性色散耗散波动方程双线性元的高精度分析

王芬玲¹|石东洋²

1.许昌学院数学与统计学院河南许昌 461000|
2.郑州大学数学与统计学院郑州 450001

收稿日期:2013-10-09 修回日期:2014-11-08 出版日期:2014-12-25 发布日期:2014-12-25
基金资助:
国家自然科学基金(10971203;11271340)、高等学校博士学科点专项科研基金(20094101110006)和河南省教育厅资助基金(14A110009)资助

High Accurary Analysis of the Bilinear Element for Nonlinear Dispersion-Dissipative Wave Equations

WANG Fen-Ling¹, SHI Dong-Yang²

1.School of Mathematics and Statistics, Xuchang University, Henan Xuchang 461000;
2.School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001

Received:2013-10-09 Revised:2014-11-08 Online:2014-12-25 Published:2014-12-25
Supported by:
国家自然科学基金(10971203;11271340)、高等学校博士学科点专项科研基金(20094101110006)和河南省教育厅资助基金(14A110009)资助

摘要/Abstract

摘要：

针对一类非线性色散耗散波动方程研究了双线性元逼近. 基于该元的高精度分析和插值后处理技巧,对于半离散格式, 在精确解的合理正则性假设下得到了$H^1$模意义下最优误差估计及超逼近性和超收敛结果. 同时, 通过构造一个新的外推格式, 导出了具有三阶精度的外推解. 最后, 建立了一个全离散逼近格式及研究其解的超逼近性.

关键词: 非线性色散耗散波动方程, 超收敛和外推, 双线性元, 半离散和全离散格式

Abstract:

The bilinear element approximation is discussed for a class of nonlinear dispersion-dissipative wave equations.
Based on the high acuraccy analysis of the element and interpolation post-processing technique, the optimal order error estimate, superclose property and superconvergence result in H¹ norm are deduced for semi-discrete scheme under the proper regularity property hypothesis of the exact solution. At the same time, the extrapolation result with three order is
obtained by constructing a new extrapolation scheme. Finally, a fully-discrete scheme is established and the
superclose property is studied.

Key words: Nonlinear dispersion-dissipative wave equations, Superconvergence and extrapolation, Bilinear element,
Semi-discrete and fully-discrete schemes

中图分类号:

65N15

王芬玲, 石东洋. 非线性色散耗散波动方程双线性元的高精度分析[J]. 数学物理学报, 2014, 34(6): 1599-1610.

WANG Fen-Ling, SHI Dong-Yang. High Accurary Analysis of the Bilinear Element for Nonlinear Dispersion-Dissipative Wave Equations[J]. Acta mathematica scientia,Series A, 2014, 34(6): 1599-1610.

参考文献

[1] 尚亚东. 方程u_tt-Δu-Δu_t-Δu_tt=f(u)的初边值问题. 应用数学学报, 2000, 23: 385--392

[2] 徐润章, 赵希人, 沈继红. 具色散耗散项的四阶波动方程解的渐近性质. 应用数学与力学, 2008, 29: 235--239

[3] Liu Y C, Xu R Z. Potential well method for Cauchy problem of generalized double dispersion equations. Journal of Mathematical Analysis Applications, 2008, 338: 1169--1187

[4] Xu R Z, Liu Y C, Yu T. Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations. Nonlinear Analysis, 2009, 71: 4977--4983

[5] 孙同军. 一类高维非线性色散耗散波动方程的有限元分析. 山东大学学报, 2003, 33: 712--716

[6] 樊明智，张建军，石东洋. 非线性色散耗散波动方程的Hermite型有限元分析. 应用数学, 2012, 25: 341--349

[7] 林群, 严宁宁. 高效有限元构造与分析. 保定: 河北大学出版社,1996

[8] Lin Q, Zhang S H, Yan N N. Asymptotic error expansion and defect correction for sobolev and viscoelasticity type equations.Journal of Computational Mathematics, 1998, 16: 51--62

[9] Lin Q, Lin J F. Extrapolation of the bilinear element approximation for Poisson equation on anisotropic meshes.
Numerical Methods for Partial Differential Equations, 2007, 23: 960--967

[10] Lin Q, Wang H, Zhang S H. Uniform optimal-order estimates for finite element methods for advection-diffusion equations. Journal of Systems Science and Complexity, 2009, 22: 555--559

[11] 林群,王凯欣, 王宏, 阴小波. 对流扩散方程双线性有限元解的一致估计. 数学的实践与认识, 2011, 41: 220--223

[12] Chen W. Eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet-Raviart scheme. Numerical Methods for Partial Differential Equations, 2005, 21: 512--520

[13] 石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推. 计算数学, 2005, 27: 369--382

[14] Lin Q, Xie H H. Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method. Applied Numerical Mathematics, 2009, 59: 1884--1893

[15] Li M X, Lin Q, Zhang S H. Extrapolation and superconvergence of the Steklov eigenvalue problem. Advance in
Computational Mathematics, 2010, 33: 25--44

[16] Lin Q, Tobiska L, Zhou A H. Superconvergence and extrapolation of nonconform low order finite elements applied to
the poisson equation, IMA Journal of Numerical Analysis, 25: 160--181

[17] Shi D Y, Mao S P, Chen S C. An anisotropic nonconforming finite element with some superconvergence results.
Journal of Computational Mathematics, 2005, 23: 261--274

[18] 石东洋, 王芬玲, 史艳华. 各向异性EQ₁^rot非协调元高精度分析的一般格式. 计算数学, 2013, 35: 239--252

[19] Shi D Y, Liang H. Superconvergence analysis Wilson element on anisotropic meshes. Applied Mathematics and Mechanics, 2007, 28: 119--125

[20] 石东洋,任金城，郝晓斌. Sobolev型方程各向异性网格下Wilson元的高精度分析. 高等学校计算数学学报，2009, 31: 169--180

[21] 石东洋,郝晓斌. Sobolev型方程各向异性Carey元的高精度分析. 工程数学学报, 2009, 26: 1021--1026

[22] 王芬玲, 石东洋. 非线性sine-Gordon方程Hermite型有限元新的超收敛分析及外推. 应用数学学报, 2012, 35: 777--788

[23] Shi D Y, Wang F L, Zhao Y M. Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations. Acta Mathematicae Applicatae Sinica, 2013, 29: 403--414

[24] Shi D Y, Zhang D. Approximation of nonconforming quasi-Wilson element for sine-Gordon equation. Journal of Computational Mathematics, 2013, 31: 271--282

[25] Shi D Y, Pei L F. Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations.
Applied Mathematics and Computation, 2013, 219: 9447--9460

[26] Zhao Y M, Wang F L, Shi D Y. Nonconforming finite element method for nonlinear dual phase lagging heat conduction equations. Acta Mathematicae Applicatae Sinica, 2013, 29: 403--414

[27] 王芬玲，石东伟，石东洋. 拟线性Sobolev方程Wilson元解的超收敛分析及外推. 工程数学学报, 2012, 29: 720--724

[28] 张铁. 发展型积分-微分方程的有限元方法. 沈阳: 东北大学出版社, 2001

[29] Lin Q, Lin J F. Finte Element Methods: Accuracy and Improvement. Beijing: Science Press, 2006