Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Acceleration of a two-grid method for eigenvalue problems


Authors: Xiaozhe Hu and Xiaoliang Cheng
Journal: Math. Comp. 80 (2011), 1287-1301
MSC (2010): Primary 65L15, 65N15, 65N25, 65N30, 65N55
DOI: https://doi.org/10.1090/S0025-5718-2011-02458-0
Published electronically: February 18, 2011
Corrigendum: Math. Comp. 84 (2015), 2701--2704.
MathSciNet review: 2785459
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides a new two-grid discretization method for solving partial differential equation or integral equation eigenvalue problems. In 2001, Xu and Zhou introduced a scheme that reduces the solution of an eigenvalue problem on a finite element grid to that of one single linear problem on the same grid together with a similar eigenvalue problem on a much coarser grid. By solving a slightly different linear problem on the fine grid, the new algorithm in this paper significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate. Numerical examples are also provided to demonstrate the efficiency of the new method.


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

  • 1. O. Axelsson and W. Layton, A two-level discretization of nonlinear boundary value problems, SIAM J. Numer. Anal. 33 (1996), no. 6, 2359–2374. MR 1427468, https://doi.org/10.1137/S0036142993247104
  • 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 89k:65132
  • 3. -, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641-787. MR 92f:65001
  • 4. F. Chatelin, Spectral approximation of linear operators, Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. MR 86d:65071
  • 5. L Chen and C.-S. Zhang, AFEM@matlab: a Matlab package of adaptive finite element methods, Tech. report, University of Maryland at College Park, 2006.
  • 6. C.-S. Chien and B.-W. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput. 27 (2006), no. 4, 1287-1304. MR 2006j:65330
  • 7. C.N. Dawson and M.F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993), Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 191-203. MR 95j:65117
  • 8. C.N. Dawson, M.F. Wheeler, and C.S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal. 35 (1998), no. 2, 435-452. MR 99b:65097
  • 9. Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
  • 10. J. Jin, S. Shu, and J. Xu, A two-grid discretization method for decoupling systems of partial differential equations, Math. Comp. 75 (2006), no. 256, 1617-1626. MR 2007e:65121
  • 11. W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput. 69 (1995), no. 2-3, 263-274. MR 95m:65191
  • 12. W. Layton and L. Tobiska, A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), no. 5, 2035-2054. MR 99g:65115
  • 13. M. Marion and J. Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal. 32 (1995), no. 4, 1170-1184. MR 96f:65136
  • 14. Beresford N. Parlett, The symmetric eigenvalue problem, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Corrected reprint of the 1980 original. MR 1490034
  • 15. G. Peters and J. H. Wilkinson, Inverse iteration, ill-conditioned equations and Newton’s method, SIAM Rev. 21 (1979), no. 3, 339–360. MR 535118, https://doi.org/10.1137/1021052
  • 16. T. Utnes, Two-grid finite element formulations of the incompressible Navier-Stokes equations, Comm. Numer. Methods Engrg. 13 (1997), no. 8, 675-684. MR 98d:76110
  • 17. J. Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal. 29 (1992), no. 2, 303-319. MR 92k:65063
  • 18. -, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231-237. MR 94m:65178
  • 19. -, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759-1777. MR 97i:65169
  • 20. J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000), no. 231, 881-909. MR 2000j:65102
  • 21. -, Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math. 14 (2001), no. 4, 293-327. MR 2002h:65198
  • 22. -, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), no. 233, 17-25. MR 2001f:65130
  • 23. -, Local and parallel finite element algorithms for eigenvalue problems, Acta Math. Appl. Sin. Engl. Ser. 18 (2002), no. 2, 185-200. MR 2004m:65184

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65L15, 65N15, 65N25, 65N30, 65N55

Retrieve articles in all journals with MSC (2010): 65L15, 65N15, 65N25, 65N30, 65N55


Additional Information

Xiaozhe Hu
Affiliation: Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou, 310027, People’s Republic of China
Email: huxiaozhezju@gmail.com

Xiaoliang Cheng
Affiliation: Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou, 310027, People’s Republic of China
Email: xiaoliangcheng@zju.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-2011-02458-0
Keywords: Eigenvalue problems, finite elements, partial differential equations, integral equations, two-grid method.
Received by editor(s): October 21, 2009
Received by editor(s) in revised form: June 15, 2010
Published electronically: February 18, 2011
Additional Notes: This work was supported in part by National Science Foundation of China (No. 10871179).
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society