Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A multi-level correction scheme for eigenvalue problems


Authors: Qun Lin and Hehu Xie
Journal: Math. Comp. 84 (2015), 71-88
MSC (2010): Primary 65N30, 65B99; Secondary 65N25, 65L15
DOI: https://doi.org/10.1090/S0025-5718-2014-02825-1
Published electronically: March 10, 2014
MathSciNet review: 3266953
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, a type of multi-level correction scheme is proposed to solve eigenvalue problems by the finite element method. This type of multi-level correction method includes multi correction steps in a sequence of finite element spaces. In each correction step, we only need to solve a source problem on a finer finite element space and an eigenvalue problem on the coarsest finite element space. The accuracy of the eigenpair approximation can be improved after each correction step. This correction scheme can improve the efficiency of solving eigenvalue problems by the finite element method.


References [Enhancements On Off] (What's this?)

  • [1] A. B. Andreev, R. D. Lazarov, and M. R. Racheva, Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems, J. Comput. Appl. Math. 182 (2005), no. 2, 333-349. MR 2147872 (2006d:65127), https://doi.org/10.1016/j.cam.2004.12.015
  • [2] I. Babuška and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275-297. MR 962210 (89k:65132), https://doi.org/10.2307/2008468
  • [3] I. Babuška and J. Osborn, Eigenvalue problems, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641-787. MR 1115240
  • [4] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258 (95f:65001)
  • [5] Françoise Chatelin, Spectral approximation of linear operators, With a foreword by P. Henrici; With solutions to exercises by Mario Ahués. Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. MR 716134 (86d:65071)
  • [6] Hongtao Chen, Shanghui Jia, and Hehu Xie, Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems, Appl. Math. 54 (2009), no. 3, 237-250. MR 2530541 (2010f:65251), https://doi.org/10.1007/s10492-009-0015-7
  • [7] Wei Chen and Qun Lin, Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method, Appl. Math. 51 (2006), no. 1, 73-88. MR 2197324 (2006k:65299), https://doi.org/10.1007/s10492-006-0006-x
  • [8] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
  • [9] John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • [10] Xiaozhe Hu and Xiaoliang Cheng, Acceleration of a two-grid method for eigenvalue problems, Math. Comp. 80 (2011), no. 275, 1287-1301. MR 2785459 (2012h:65264), https://doi.org/10.1090/S0025-5718-2011-02458-0
  • [11] Shanghui Jia, Hehu Xie, Xiaobo Yin, and Shaoqin Gao, Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods, Appl. Math. 54 (2009), no. 1, 1-15. MR 2476018 (2010b:35366), https://doi.org/10.1007/s10492-009-0001-0
  • [12] Qun Lin, Some problems concerning approximate solutions of operator equations, Acta Math. Sinica 22 (1979), no. 2, 219-230 (Chinese, with English summary). MR 542459 (80k:65056)
  • [13] Qun Lin, Hung-Tsai Huang, and Zi-Cai Li, New expansions of numerical eigenvalues for $ -\Delta u=\lambda \rho u$ by nonconforming elements, Math. Comp. 77 (2008), no. 264, 2061-2084. MR 2429874 (2010e:65202), https://doi.org/10.1090/S0025-5718-08-02098-X
  • [14] Q. Lin and J. Lin, Finite element methods: Accuracy and improvement, China Sci. Tech. Press, 2005.
  • [15] Qun Lin and Tao Lü, Asymptotic expansions for finite element eigenvalues and finite element solution, Extrapolation procedures in the finite element method (Bonn, 1983), Bonner Math. Schriften, vol. 158, Univ. Bonn, Bonn, 1984, pp. 1-10. MR 793412 (86j:65146)
  • [16] Qun Lin and Hehu Xie, Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method, Appl. Numer. Math. 59 (2009), no. 8, 1884-1893. MR 2536089 (2010f:65259), https://doi.org/10.1016/j.apnum.2009.01.011
  • [17] Q. Lin and N. Yan, The construction and analysis of high efficiency finite element methods, Hebei University Publishers, 1995.
  • [18] Ahmed Naga, Zhimin Zhang, and Aihui Zhou, Enhancing eigenvalue approximation by gradient recovery, SIAM J. Sci. Comput. 28 (2006), no. 4, 1289-1300 (electronic). MR 2255457 (2007k:65169), https://doi.org/10.1137/050640588
  • [19] Milena R. Racheva and Andrey B. Andreev, Superconvergence postprocessing for eigenvalues, Comput. Methods Appl. Math. 2 (2002), no. 2, 171-185. MR 1930846 (2003k:65139)
  • [20] Y. Saad, Numerical Methods for Large Eigenvalue Problems-2nd Edition, SIAM, 2011.
  • [21] Haijun Wu and Zhimin Zhang, Enhancing eigenvalue approximation by gradient recovery on adaptive meshes, IMA J. Numer. Anal. 29 (2009), no. 4, 1008-1022. MR 2557054 (2010k:65249), https://doi.org/10.1093/imanum/drn050
  • [22] Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581-613. MR 1193013 (93k:65029), https://doi.org/10.1137/1034116
  • [23] Jinchao Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal. 29 (1992), no. 2, 303-319. MR 1154268 (92k:65063), https://doi.org/10.1137/0729020
  • [24] Jinchao Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231-237. MR 1257166 (94m:65178), https://doi.org/10.1137/0915016
  • [25] Jinchao Xu and Aihui Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), no. 233, 17-25. MR 1677419 (2001f:65130), https://doi.org/10.1090/S0025-5718-99-01180-1
  • [26] Jinchao Xu and Aihui Zhou, Local and parallel finite element algorithms for eigenvalue problems, Acta Math. Appl. Sin. Engl. Ser. 18 (2002), no. 2, 185-200. MR 2008551 (2004m:65184), https://doi.org/10.1007/s102550200018
  • [27] Aihui Zhou, Multi-level adaptive corrections in finite dimensional approximations, J. Comput. Math. 28 (2010), no. 1, 45-54. MR 2603580 (2011b:65084), https://doi.org/10.4208/jcm.2009.09-m1003

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65B99, 65N25, 65L15

Retrieve articles in all journals with MSC (2010): 65N30, 65B99, 65N25, 65L15


Additional Information

Qun Lin
Affiliation: LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Email: linq@lsec.cc.ac.cn

Hehu Xie
Affiliation: LSEC, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Email: hhxie@lsec.cc.ac.cn

DOI: https://doi.org/10.1090/S0025-5718-2014-02825-1
Keywords: Eigenvalue problem, multi-level correction scheme, finite element method, multi-space, multi-grid
Received by editor(s): January 23, 2011
Received by editor(s) in revised form: December 20, 2011, October 9, 2012, and April 18, 2013
Published electronically: March 10, 2014
Additional Notes: This work was supported in part by the National Science Foundation of China (NSFC 11001259, 11031006, 11371026, 11201501, 2011CB309703), the Croucher Foundation of the Hong Kong Baptist University and the National Center for Mathematics and Interdisciplinary Science, CAS and the President Foundation of AMSS-CAS
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society