计算数学
       首页 |  期刊介绍 |  编委会 |  投稿指南 |  期刊订阅 |  下载中心 |  留言板 |  联系我们 |  在线办公 | 
计算数学  2018, Vol. 40 Issue (2): 149-170    DOI:
论文 最新目录 | 下期目录 | 过刊浏览 | 高级检索 Previous Articles  |  Next Articles  
高波数问题的超收敛性
杜宇
湘潭大学数学与计算科学学院, 湘潭 411105
SUPERCONVERGENCE ANALYSIS FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER
Du Yu
Department of Mathematics, Xiangtan University, Xiangtan 411105, China
 全文: PDF (784 KB)   HTML (1 KB)   输出: BibTeX | EndNote (RIS)      背景资料
摘要 本文考虑求解Helmholtz方程的有限元方法的超逼近性质以及基于PPR后处理方法的超收敛性质.我们首先给出了矩形网格上的p-次元在收敛条件kkh2p+1C0下的有限元解和基于Lobatto点的有限元插值之间的超逼近以及重构的有限元梯度和精确解之间的超收敛分析.然后我们给出了四边形网格上的线性有限元方法的分析.这些估计都给出了与波数k和网格尺寸h的依赖关系.同时我们回顾了三角形网格上的线性有限元的超收敛结果.最后我们给出了数值实验并且结合Richardson外推进一步减少了误差.
服务
把本文推荐给朋友
加入我的书架
加入引用管理器
E-mail Alert
RSS
作者相关文章
关键词亥姆霍兹方程   PPR方法   超收敛   超逼近     
Abstract: We study the supercloseness property of the finite element methods and their superconvergence behavior after post-processed by the polynomial preserving recovery (PPR) on both Cartesian and quadrilateral meshes for the two dimensional Helmholtz equation. The error estimate with explicit dependence on the wave number k and the mesh condition parameter α is derived. We first analyze the supercloseness between the finite element solution and the interpolation and the superconvergence for the recovered gradient by PPR under the assumption k(kh)2p+1C0(h is the mesh size) on Cartesian meshes. We then analyze the supercloseness and superconvergence for the linear finite element method on quadrilateral meshes. We also recall our work about superconvergence property of the linear FEM on triangle meshes. Furthermore, we estimate the error between the numerical gradient and recovered gradient, which motivate us to define the a posteriori error estimator and design a Richardson extrapolation to post-process the recovered gradient by PPR. Finally, Some numerical examples are provided to confirm the theoretical results of superconvergence analysis.
Key wordsHelmholtz equation   large wave number   superconvergence   PPR   finite element methods   
收稿日期: 2017-08-26;
基金资助:

国家自然科学基金青年科学基金(11601026).

引用本文:   
. 高波数问题的超收敛性[J]. 计算数学, 2018, 40(2): 149-170.
. SUPERCONVERGENCE ANALYSIS FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER[J]. Mathematica Numerica Sinica, 2018, 40(2): 149-170.
 
[1] Wahlbin L. Superconvergence in Galerkin finite element methods[J]. Springer, 2006.
[2] Chen C, Hu S. The highest order superconvergence for bi-k degree rectangular elements at nodes:a proof of 2k-conjecture[J]. Mathematics of Computation, 2013, 82(283):1337-1355.
[3] Douglas J. Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems[J]. Topics in numerical analysis, 1973, pages 89-92.
[4] Douglas J. Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces[J]. Numerische Mathematik, 1974, 22(2):99-109.
[5] Naga A, Zhang Z. A posteriori error estimates based on the polynomial preserving recovery[J]. SIAM J. Numer. Anal., 2004, 42:1780-1800.
[6] Naga A, Zhang Z. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D[J]. Discrete and continuous dynamical systems series B, August 2005, 5(3):759-798.
[7] Zhang Z, Naga A. A new finite element gradient recovery method:Superconvergence property[J]. SIAM J. Sci. Comput., 2005, 26:1192-1213.
[8] Zienkiewicz O, Zhu J. A simple error estimator and adaptive procedure for practical engineerng analysis[J]. International journal for numerical methods in engineering, 1987, 24(2):337-357.
[9] Zienkiewicz O, Zhu J. The superconvergent patch recovery and a posteriori error estimates. Part 1:The recovery technique[J]. International Journal for Numerical Methods in Engineering, 1992,33(7):1331-1364.
[10] Zienkiewicz O, J.Z. Z. The superconvergent patch recovery and a posteriori error estimates. Part 2:Error estimates and adaptivity[J]. International Journal for Numerical Methods in Engineering, 1992, 33(7):1365-1382.
[11] Ainsworth M, Oden J. A posteriori error estimation in finite element analysis[J]. John Wiley & Sons, 2011.
[12] Babuška I, Strouboulis T, Upadhyay C. Validation of a posteriori error estimators by numerical approach[J]. International journal for numerical methods in engineering, 1994, 37(7):1073-1123.
[13] Carstensen C, Bartels S. Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I:Low order conforming, nonconforming, and mixed FEM[J]. Mathematics of Computation, 2002, 71(239):945-969.
[14] Guo H, Zhang Z, Zhao R. Superconvergent two-grid methods for elliptic eigenvalue problems[J]. Journal of Scientific Computing, 2017, 70(1):125-148.
[15] Wu H, Zhang Z. Enhancing eigenvalue approximation by gradient recovery on adaptive meshes[J]. IMA journal of numerical analysis, 2008, 29(4):1008-1022.
[16] Zhu L, Wu H. Pre-asymptotic error Analysis of CIP-FEM and FEM for Helmholtz Equation with high Wave Number. Part Ⅱ:hp version[J]. SIAM J. Numer. Anal., 2013, 51(3):1828-1852.
[17] Du Y, Wu H. Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2015, 53(2):782-804.
[18] Aziz A, Kellogg R. A scattering problem for the Helmholtz equation[C]. In Advances in Computer Methods for Partial Differential Equations-Ⅲ, volume 1, pages 93-95, 1979.
[19] Jr J D, Santos J, Sheen D. Approximation of scalar waves in the space-frequency domain[J]. Math. Models Methods Appl. Sci., 1994, 4:509-531.
[20] Schatz A. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[J]. Math. Comp., 1974, 28:959-962.
[21] Melenk J M, Sauter S. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[J]. Math. Comp., 2010, 79(272):1871-1914.
[22] Melenk J M, Sauter S. Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation[J]. SIAM J. Numer. Anal., 2011, 49(3):1210-1243.
[23] Wu H. Pre-asymptotic error Analysis of CIP-FEM and FEM for Helmholtz Equation with high Wave Number. Part I:Linear version[J]. IMA J. Numer. Anal., 2014, 34:1266-1288.
[24] Haldenwang P, Labrosse G, Abboudi S, Deville M. Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation[J]. Journal of Computational Physics, 1984, 55(1):115-128.
[25] Feng X, Wu H. Discontinuous Galerkin methods for the Helmholtz equation with large wave numbers[J]. SIAM J. Numer. Anal., 2009, 47(4):2872-2896.
[26] Feng X, Wu H. hp-discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number[J]. Math. Comp., 2011, 80(276):1997-2024.
[27] Zhu L, Du Y. Pre-asymptotic error analysis of hp-Interior Penalty Discontinuous Galerkin methods for the Helmholtz Equation with large wave number[J]. Comput. Math. Appl., 2015, 70:917-933.
[28] Du Y, Zhu L. Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number[J]. J. Sci. Comput., April 2016, 67(1):130-152.
[29] Chen Z, Xiang X. A Source Transfer Domain Decomposition Method For Helmholtz Equations in Unbounded Domain[J]. SIAM J. Numer. Anal., 2013, 51:2331-2356.
[30] Du Y, Wu H, Zhang Z. Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number[J]. arXiv:1703.00156, 2017.
[31] Brenner S, Scott L. The mathematical theory of finite element methods[M]. Springer, New York, third edition, 2008.
[32] Ciarlet P G. The finite element method for elliptic problems[M]. North-Holland Pub. Co., New York, 1978.
[33] Melenk J, Parsania A, Sauter S. General DG-Methods for Highly Indefinite Helmholtz Problems[J]. Journal of Scientific Computing, 2013, 57:536-581.
[34] Zhang Z. Polynomial preserving recovery for anisotropic and irregular grids[J]. J. Comput. Math., 2004, 22:331-340.
[35] Guo H, Yang X. Polynomial preserving recovery for high frequency wave propagation[J]. Journal of Scientific Computing, 2017, 71(2):594-614.
[36] Zhang T, Yu S. The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method[J]. Applied Numerical Mathematics, 2014, 85:128-141.
[37] Acosta G, Durán R. Error Estimates for Q1 Isoparametric Elements Satisfying a Weak Angle Condition[J]. SIAM Journal on Numerical Analysis, 2000, 38(4):1073-1088.
[38] Zhang Z. Polynomial preserving gradient recovery and a posteriori estimate for bilinear element on irregular quadrilaterals[J]. Internat. J. Numer. Anal. Model., 2004, 1:1-24.
[39] Marchuk G, Shaidurov V. Difference Methods and Their Extrapolation[M]. Springer-Verlag, New York, 1983.
[40] Blum H, Rannacher R. Asymptotic error expansion and Richardson extrapolation for linear finite elements[J]. Numer. Math., 1986, 49:11-38.
[41] Wang J. Asymptotic expansions and L-error estimates for mixed finite element methods for second order elliptic problems[J]. Numer. Math., 1989, 55:401-430.
[42] Helfrich P. Asymptotic expansion for the finite element approximations of parabolic problems[J]. Bonner Math. Schriften, 1983, 158:11-30.
[43] Lin Q, Zhang S, Yan N. Asymptotic error expansion and defect correction for Sobolev and viscoelasticity type equations[J]. J. Comput. Math., 1998, 16:57-62.
[1] 许秀秀, 黄秋梅. 拟等级网格下非线性延迟微分方程间断有限元法[J]. 计算数学, 2016, 38(3): 281-288.
[2] 石东洋, 张厚超, 王瑜. 一类非线性四阶双曲方程扩展的混合元方法的超收敛分析[J]. 计算数学, 2016, 38(1): 65-82.
[3] 赵艳敏, 石东洋, 王芬玲. 非线性Schrödinger方程新混合元方法的高精度分析[J]. 计算数学, 2015, 37(2): 162-178.
[4] 石东洋, 王芬玲, 樊明智, 赵艳敏. sine-Gordon方程的最低阶各向异性混合元高精度分析新途径[J]. 计算数学, 2015, 37(2): 148-161.
[5] 石东洋, 史艳华, 王芬玲. 四阶抛物方程H1-Galerkin混合有限元方法的超逼近及最优误差估计[J]. 计算数学, 2014, 36(4): 363-380.
[6] 石东洋, 王芬玲, 赵艳敏. 非线性sine-Gordon方程的各向异性线性元高精度分析新模式[J]. 计算数学, 2014, 36(3): 245-256.
[7] 石东洋, 张亚东. 抛物型方程一个新的非协调混合元超收敛性分析及外推[J]. 计算数学, 2013, 35(4): 337-352.
[8] 尹云辉, 祝鹏, 杨宇博. 奇异摄动问题在Bakhvalov-Shishkin网格上的有限元超收敛[J]. 计算数学, 2013, 34(4): 257-265.
[9] 石东洋, 王芬玲, 史艳华. 各向异性EQ1rot非协调元高精度分析的一般格式[J]. 计算数学, 2013, 35(3): 239-252.
[10] 石东洋, 唐启立, 董晓靖. 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.
[11] 石东洋, 谢萍丽, 于志云. 各向异性网格下的双三次Hermite元的超逼近分析[J]. 计算数学, 2008, 30(4): 337-348.
[12] 李郴良,陈传淼,许学军,. 基于超收敛和外推方法的一类新的瀑布型多重网格方法[J]. 计算数学, 2007, 29(4): 439-448.
[13] 李焕荣,罗振东,李潜,. 二维粘弹性问题的广义差分法及其数值模拟[J]. 计算数学, 2007, 29(3): 251-262.
[14] 喻海元,黄云清,. 变系数情形下Criss-Cross三角形线性元的渐近展式与超收敛[J]. 计算数学, 2007, 29(3): 325-336.
[15] 石东洋,汪松玉,陈绍春,. 两类各向异性非协调元的某些超收敛性质分析[J]. 计算数学, 2007, 29(3): 263-272.

Copyright 2008 计算数学 版权所有
中国科学院数学与系统科学研究院 《计算数学》编辑部
北京2719信箱 (100190) Email: gxy@lsec.cc.ac.cn
本系统由北京玛格泰克科技发展有限公司设计开发
技术支持: 010-62662699 E-mail:support@magtech.com.cn
京ICP备05002806号-10