A multilevel Newton's method for eigenvalue problems

Yunhui He; Yu Li; Hehu Xie; Chun'guang You; Ning Zhang

Applications of Mathematics (2018)

  • Volume: 63, Issue: 3, page 281-303
  • ISSN: 0862-7940

Abstract

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We propose a new type of multilevel method for solving eigenvalue problems based on Newton's method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.

How to cite

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He, Yunhui, et al. "A multilevel Newton's method for eigenvalue problems." Applications of Mathematics 63.3 (2018): 281-303. <http://eudml.org/doc/294519>.

@article{He2018,
abstract = {We propose a new type of multilevel method for solving eigenvalue problems based on Newton's method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.},
author = {He, Yunhui, Li, Yu, Xie, Hehu, You, Chun'guang, Zhang, Ning},
journal = {Applications of Mathematics},
keywords = {eigenvalue problem; finite element method; Newton's method; multilevel iteration},
language = {eng},
number = {3},
pages = {281-303},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A multilevel Newton's method for eigenvalue problems},
url = {http://eudml.org/doc/294519},
volume = {63},
year = {2018},
}

TY - JOUR
AU - He, Yunhui
AU - Li, Yu
AU - Xie, Hehu
AU - You, Chun'guang
AU - Zhang, Ning
TI - A multilevel Newton's method for eigenvalue problems
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 281
EP - 303
AB - We propose a new type of multilevel method for solving eigenvalue problems based on Newton's method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.
LA - eng
KW - eigenvalue problem; finite element method; Newton's method; multilevel iteration
UR - http://eudml.org/doc/294519
ER -

References

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  1. Babuška, I., Osborn, J. E., 10.2307/2008468, Math. Comput. 52 (1989), 275-297. (1989) Zbl0675.65108MR0962210DOI10.2307/2008468
  2. Babuška, I., Osborn, J., Eigenvalue problems, Handbook of Numerical Analysis. II: Finite Element Methods (Part 1) North-Holland, Amsterdam P. G. Ciarlet, J. L. Lions (1991), 641-787. (1991) Zbl0875.65087MR1115240
  3. Brandt, A., McCormick, S., Ruge, J., 10.1137/0904019, SIAM J. Sci. Stat. Comput. 4 (1983), 244-260. (1983) Zbl0517.65083MR0697178DOI10.1137/0904019
  4. Brezzi, F., Fortin, M., 10.1007/978-1-4612-3172-1, Springer Series in Computational Mathematics 15, Springer, New York (1991). (1991) Zbl0788.73002MR1115205DOI10.1007/978-1-4612-3172-1
  5. Chatelin, F., 10.1137/1.9781611970678, Computer Science and Applied Mathematics, Academic Press, New York (1983). (1983) Zbl0517.65036MR0716134DOI10.1137/1.9781611970678
  6. Davidson, E. R., Thompson, W. J., 10.1063/1.4823212, Comput. Phys. 7 (1993), 519-522. (1993) DOI10.1063/1.4823212
  7. Durán, R. G., Padra, C., Rodríguez, R., 10.1142/S0218202503002878, Math. Models Methods Appl. Sci. 13 (2003), 1219-1229. (2003) Zbl1072.65144MR1998821DOI10.1142/S0218202503002878
  8. Golub, G. H., Loan, C. F. Van, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore (2013). (2013) Zbl1268.65037MR3024913
  9. Hackbusch, W., 10.1137/0716015, SIAM J. Numer. Anal. 16 (1979), 201-215. (1979) Zbl0403.65043MR0526484DOI10.1137/0716015
  10. Hackbusch, W., 10.1007/978-3-662-02427-0, Springer Series in Computational Mathematics 4, Springer, Berlin (1985). (1985) Zbl0595.65106MR0814495DOI10.1007/978-3-662-02427-0
  11. Kressner, D., 10.1007/s00211-009-0259-x, Numer. Math. 114 (2009), 355-372. (2009) Zbl1191.65054MR2563153DOI10.1007/s00211-009-0259-x
  12. Larson, M. G., 10.1137/S0036142997320164, SIAM J. Numer. Anal. 38 (2000), 608-625. (2000) Zbl0974.65100MR1770064DOI10.1137/S0036142997320164
  13. Lin, Q., Lin, J., Finite Element Methods: Accuracy and Improvement, Mathematics Monograph Series 1, Elsevier (2006). (2006) 
  14. Lin, Q., Xie, H., An observation on the Aubin-Nitsche lemma and its applications, Math. Pract. Theory 41 Chinese (2011), 247-258. (2011) Zbl1265.65235MR2931490
  15. Lin, Q., Xie, H., A type of multigrid method for eigenvalue problem, Technical report, Research Report of ICM-SEC, 2011 Available at http://www.cc.ac.cn/2011researchreport/201106.pdf. 
  16. Lin, Q., Xie, H., A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems, Proc. Int. Conf. Applications of Mathematics 2012 J. Brandts et al. Academy of Sciences of the Czech Republic, Institute of Mathematics, Praha (2012), 134-143. (2012) Zbl1313.65298MR3204407
  17. Lin, Q., Yan, N., The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Publishers, Shijiazhuang (1996). (1996) 
  18. Saad, Y., 10.1137/1.9781611970739, Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester; Halsted Press, New York (1992). (1992) Zbl0991.65039MR1177405DOI10.1137/1.9781611970739
  19. Shaidurov, V. V., 10.1007/978-94-015-8527-9, Mathematics and Its Applications 318, Kluwer Academic Publishers Group, Dordrecht (1995). (1995) Zbl0837.65118MR1335921DOI10.1007/978-94-015-8527-9
  20. Sleijpen, G. L. G., Vorst, H. A. Van der, 10.1137/S0895479894270427, SIAM J. Matrix Anal. Appl. 17 (1996), 401-425. (1996) Zbl0860.65023MR1384515DOI10.1137/S0895479894270427
  21. Sleijpen, G. L. G., Vorst, H. A. van der, The Jacobi-Davidson method for eigenvalue problems and its relation with accelerated inexact Newton scheme, IMACS 1996: Iterative Methods in Linear Algebra II S. Margenov, P. Vassilevski Blagoevgrad, Bulgaria (1996). (1996) 
  22. Sleijpen, G. L. G., Vorst, H. A. Van der, 10.1137/S0036144599363084, SIAM Rev. 42 (1998), 267-293. (1998) Zbl0949.65028MR1778354DOI10.1137/S0036144599363084
  23. Xie, H., 10.1016/j.jcp.2014.06.030, J. Comput. Phys. 274 (2014), 550-561. (2014) Zbl1352.65631MR3231782DOI10.1016/j.jcp.2014.06.030
  24. Xie, H., 10.1093/imanum/drt009, IMA J. Numer. Anal. 34 (2014), 592-608. (2014) Zbl1312.65178MR3194801DOI10.1093/imanum/drt009
  25. Xu, J., 10.1137/1034116, SIAM Rev. 34 (1992), 581-613. (1992) Zbl0788.65037MR1193013DOI10.1137/1034116
  26. Xu, J., Zhou, A., 10.1090/S0025-5718-99-01180-1, Math. Comput. 70 (2001), 17-25. (2001) Zbl0959.65119MR1677419DOI10.1090/S0025-5718-99-01180-1

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