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): 119-134    DOI:
论文 最新目录 | 下期目录 | 过刊浏览 | 高级检索 Previous Articles  |  Next Articles  
高波数波动问题的多水平方法
卢培培1, 许学军2,3
1. 苏州大学数学科学学院, 苏州 215006;
2. LSEC, 中国科学院数学与系统科学研究院, 北京 100190;
3. 同济大学数学科学学院, 上海, 200092
MULTILEVEL METHODS FOR THE WAVE PROBLEMS WITH HIGH WAVE NUMBER
Lu Peipei1, Xu Xuejun2,3
1. School of Mathematics Sciences, Soochow University, Suzhou 215006, China;
2. LSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China;
3. School of Mathematical Sciences, Tongji University, Shanghai 200092, China
 全文: PDF (585 KB)   HTML (1 KB)   输出: BibTeX | EndNote (RIS)      背景资料
摘要 本文主要讨论求解高波数Helmholtz方程的多水平方法,主要回顾了一些具有代表性的多重网格方法.如Erlangga等人的shifted Laplacian预处理的多重网格法;Elman等提出的修正的多重网格方法;以及我们的基于连续内罚有限元(CIP-FEM)离散代数系统的多水平算法.最后还介绍了求解高波数时谐Maxwell方程的CIP-FEM离散代数系统的多水平算法.
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
关键词高波数   Helmholtz方程   时谐Maxwell方程   多水平方法     
Abstract: In this paper, we mainly review some multilevel preconditioners for the Helmholtz equation with high wave number, which include the shifted Laplacian preconditioner proposed by Erlangga etc.; A modified multigrid method proposed by Elman etc.; and our multilevel methods based on the continuous interior penalty method(CIP-FEM). Finally, we also introduce an efficient multilevel method for the time-harmonic Maxwell equation with high wave number.
Key wordshigh wave number   Helmholtz equation   time-harmonic Maxwell equation   multilevel method   
收稿日期: 2017-08-09;
基金资助:

国家自然科学基金(11401417,11671302)资助项目.

引用本文:   
. 高波数波动问题的多水平方法[J]. 计算数学, 2018, 40(2): 119-134.
. MULTILEVEL METHODS FOR THE WAVE PROBLEMS WITH HIGH WAVE NUMBER[J]. Mathematica Numerica Sinica, 2018, 40(2): 119-134.
 
[1] Adams R. Sobolev Spaces. Academic Press. New York, 1975.
[2] Babuška I and Sauter S A. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?[J]. SIAM Rev., 2000, 42:451-484.
[3] Bayliss A, Goldstein C, Turkel E. An iterative method for Helmholtz equation[J]. J Comput Phys., 1983, 49:443-457.
[4] Brandt A and Livshits I. Wave-ray multigrid method for standing wave equations[J]. Electron. Trans. Numer. Anal., 1997, 6:162-181.
[5] Chen H, Wu H and Xu X. Multilevel preconditioner with stable coarse grid corrections for the Helmholtz equation[J]. SIAM J. Sci. Comput., 2015, 37:A221-A244.
[6] Elman H C, Ernst O G and O'Leary D P. A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations[J]. SIAM J. Sci. Comput., 2001, 23:1291-1315.
[7] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:hierarchical matrix representation[J]. Comm. Pure Appl. Math., 2011, 64:697-735.
[8] Engquist B and Ying L. Sweeping preconditioner for the Helmholtz equation:moving perfectly matched layers[J]. Multiscale Model. Simul., 2011, 9:686-710.
[9] Erlangga Y A. Advances in iterative methods and preconditioners for the Helmholtz equation[J]. Arch. Comput. Methods Eng., 2008, 15:37-66.
[10] Erlangga Y A, Vuik C and Oosterlee C W. On a class of preconditioners for solving the Helmholtz equation[J]. Appl. Numer. Math., 2004, 50:409-425.
[11] Erlangga Y A, Oosterlee C W and Vuik C. A novel multigrid based preconditioner for heterogeneous Helmholtz problems[J]. SIAM J. Sci. Comput., 2006, 27:1471-1492.
[12] Ernst O G and Gander M J. Why it is difficult to solve Helmholtz problems with classical iterative methods. Numerical Analysis of Multiscale Problems, Durham LMS Symposium, Citeseer, 2010.
[13] Feng X and Wu H. Discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. SIAM J. Numer. Anal., 2009, 47:2872-2896.
[14] Feng X and Wu H. hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. Math. Comp., 2011, 80:1997-2024.
[15] Ihlenburg F. Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York, 1998.
[16] Laird A, Giles M. Preconditioned iterative solution of the 2D Helmholtz equation, Technical Report NA 02-12, Comp Lab, Oxford Univ, 2002.
[17] Livshits I and Brandt A. Accuracy properties of the wave-ray multigrid algorithm for Helmholtz equations[J]. SIAM J. Sci. Comput., 2006, 28:1228-1251.
[18] Lu P and Xu X. A robust multilevel method for the time-harmonic Maxwell equation with high wave number[J]. SIAM J. Sci. Comput., 2016, 38:A856-A874.
[19] Lu P, Wu H and Xu X. Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number, submitted.
[20] Melenk J M and Sauter S. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[M]. Math. Comp., 2010, 79:1871-1914.
[21] Melenk J M and Sauter S. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation[J]. SIAM J. Numer. Anal., 2011, 49:1210-1243.
[22] Starke G. On the Performance of Krylov Subspace Iterations as Smoothers in Multigrid Methods. manuscript, 1995.
[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] Zhu L and 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., 2012, 51:1828-1852.
[25] Saad Y. Iterative Methods for Sparse Linear Systems. PWS, Boston, 1995.
[1] 王坤, 张扬, 郭瑞. Helmholtz方程有限差分方法概述[J]. 计算数学, 2018, 40(2): 171-190.
[2] 武海军. 高波数Helmholtz方程的有限元方法和连续内罚有限元方法[J]. 计算数学, 2018, 40(2): 191-213.
[3] 骆其伦, 黎稳. 二维Helmholtz方程的联合紧致差分离散方程组的预处理方法[J]. 计算数学, 2017, 39(4): 407-420.
[4] 郑权, 高玥, 秦凤. Helmholtz方程外边值问题的基于修正的DtN边界条件的有限元方法[J]. 计算数学, 2016, 38(2): 200-211.
[5] 袁龙, 胡齐芽. 复波数Helmholtz方程和时谐Maxwell方程组的平面波间断Petrov-Galerkin方法[J]. 计算数学, 2015, 36(3): 185-196.
[6] 孟文辉, 王连堂. Helmholtz方程周期Green函数及其偏导数截断误差收敛阶的分析[J]. 计算数学, 2015, 37(2): 123-136.
[7] 柯日焕, 黎稳. 用CCD法离散求解二维Helmholtz方程的数值方法[J]. 计算数学, 2013, 34(3): 221-230.
[8] 段艳婷, 王连堂, 徐建丽. 二维Helmholtz方程外问题的数值解法[J]. 计算数学, 2011, 32(1): 57-63.
[9] 张敏,杜其奎,. 椭圆外区域上Helmholtz问题的自然边界元法[J]. 计算数学, 2008, 30(1): 75-88.
[10] 杨超,孙家昶. 一类六边形网格上拉普拉斯4点差分格式及其预条件子[J]. 计算数学, 2005, 27(4): 437-448.
[11] 贾祖朋,邬吉明,余德浩. 三维Helmholtz方程外问题的自然边界元与有限元耦合法[J]. 计算数学, 2001, 23(3): 357-368.
[12] 余德浩,贾祖朋. 二维Helmholtz方程外问题基于自然边界归化的非重叠型区域分解算法[J]. 计算数学, 2000, 22(2): 227-240.

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