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分层网格上奇异摄动问题的一致NIPG分析

杨宇博1, 祝鹏2, 尹云辉2   

  1. 1. 嘉兴学院 南湖学院, 浙江嘉兴 314001;
    2. 嘉兴学院 数理与信息工程学院, 浙江嘉兴 314001
  • 收稿日期:2013-12-26 出版日期:2014-11-15 发布日期:2014-12-06
  • 基金资助:

    浙江省自然科学基金(LQ12A01014)和浙江省教育厅科研项目(Y201330020)资助项目.

杨宇博, 祝鹏, 尹云辉. 分层网格上奇异摄动问题的一致NIPG分析[J]. 计算数学, 2014, 36(4): 437-448.

Yang Yubo, Zhu Peng, Yin Yunhui. UNIFORM ANALYSIS OF THE NIPG METHOD ON GRADED MESHES FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS[J]. Mathematica Numerica Sinica, 2014, 36(4): 437-448.

UNIFORM ANALYSIS OF THE NIPG METHOD ON GRADED MESHES FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS

Yang Yubo1, Zhu Peng2, Yin Yunhui2   

  1. 1. Nanhu College, Jiaxing University, Jiaxing 314001, Zhejiang, China;
    2. School of Math-Physics and Information Engineering, Jiaxing University, Jiaxing 314001, Zhejiang, China
  • Received:2013-12-26 Online:2014-11-15 Published:2014-12-06
本文采用非对称内罚间断有限元方法(以下简称 NIPG 方法)求解一维对流扩散型奇异摄动问题. 理论上证明了采用拉格朗日线性元的 NIPG 方法在分层网格上至多相差一个关于摄动参数对数因子的拟最优阶的一致收敛性, 即在能量范数度量下其误差估计为O(log2(1/ε)/N), 其中N为网格剖分中单元个数. 数值算例验证了理论分析的正确性.
A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for one-dimensional singularly perturbed convection-diffusion problem. On graded meshes with Lagrange linear elements, the method is shown to be convergent, uniformly in the perturbation parameter εL, of optimal error O(log2(1/ε)/N) in the energy norm, up to a logarithmic factor, where N is the number of mesh. Finally, through numerical experiments, the authors verified the theoretical result.

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[1] Roos H G, Stynes M, Tobiska L. Robust numerical methods for singularly perturbed differential equations[M]. Springer Series in Computational Mathematics, Volume 24, Springer-Verlag Berlin Heidelberg, 2008.

[2] Linss T. Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection diffusion problem[J]. IMA J. Numer. Anal., 2000, 20(4): 621-632.

[3] Linss T. Layer-adapted meshes for convection-diffusion problems[J]. Comp. Meth. Appl. Mech. Engng, 2003, 192(9): 1061-1105.

[4] Stynes M, O'Riorddan E. A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem[J]. J. Math. Anal. Appl., 1997, 214(1): 36-54.

[4] Linss T, Stynes M. Numerical methods on Shishkin meshes for linear convection-diffusion problems[J]. Comp. Meth. Appl. Mech. Engng, 2001,190(28): 3527-3542.

[6] Stynes M, Tobiska L. The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy[J]. SIAM. J. Numer. Anal., 2003, 41: 1620-1642.

[7] 祝鹏, 尹云辉, 杨宇博. 奇异摄动问题内罚间断有限方法的 最优阶一致收敛性分析[J]. 计算数学, 2013, 35(3): 323-336.

[8] 尹云辉, 祝鹏, 杨宇博. 奇异摄动问题 在Bakhvalov-Shishkin 网格上的流线扩散有限元逼近[J]. 计算数学, 2013, 35(4): 365-376.

[9] Durán R G, Lombardi A L. Finite element approximation of convection diffusion problems using graded meshes[J]. Appl. Numer. Math., 2006, 56: 1314-1325.

[10] Guoqing Zhu, Shaochun Chen. Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems[J]. J. Comput. Appl. Math., 2010, 234: 3048-3063.

[11] Linss T. The necessity of Shishkin decompositions[J]. Appl. Math. Lett. , 2001, 14(7): 891-896.

[12] Zhu P, Xie Z Q, Zhou S Z. A coupled continuous-discontinuous FEM approach for convection diffusion equations[J]. Acta Math. Sci., 2011, 31B: 601-612.

[13] 李立康, 郭毓(马匋). 索伯列夫空间引论[M]. 上海: 上海科学技术出版社, 1981.

[14] Ciarlet P G. The finite element method for elliptic problems[M]. Amsterdam: North-Holland, 1978.
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