Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

Request Permissions   Purchase Content 


The second order spectrum and optimal convergence

Author: Michael Strauss
Journal: Math. Comp. 82 (2013), 2305-2325
MSC (2010): Primary 47A75, 47B15
Published electronically: April 9, 2013
MathSciNet review: 3073202
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces. The convergence to eigenspaces is new, while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude.

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

  • 1. D. BOFFI, F. BREZZI, L. GASTALDI, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (1999) 121-140. MR 1642801 (2000i:65175)
  • 2. D. BOFFI, R. G. DURAN, AND L. GASTALDI. A remark on spurious eigenvalues in a square. Appl. Math. Lett. 12(3) (1999) 107-114. MR 1749719 (2000m:65133)
  • 3. L. BOULTON, Non-variational approximation of discrete eigenvalues of self-adjoint operators. IMA J. Numer. Anal. 27 (2007) 102-112. MR 2289273 (2007m:47034)
  • 4. L. BOULTON, Limiting set of second order spectra. Math. Comp. 75 (2006) 1367-1382. MR 2219033 (2007h:47049)
  • 5. L. BOULTON, M. LEVITIN, On approximation of the eigenvalues of perturbed periodic Schrodinger operators, J. Phys. A: Math. Theor. 40 (2007), 9319-9329. MR 2345295 (2008e:35033)
  • 6. L. BOULTON, M. STRAUSS, On the convergence of second-order spectra and multiplicity. Proc. R. Soc. A 467 (2011) 264-284. MR 2764683
  • 7. L. BOULTON, N. BOUSSAID, Non-variational computation of the eigenstates of dirac operators with radially symmetric potentials. LMS J. Comput. Math. 13 (2010) 10-32. MR 2593910 (2011i:65195)
  • 8. F. CHATELIN, Spectral Approximation of Linear Operators. Academic Press (1983). MR 716134 (86d:65071)
  • 9. M. DAUGE, M. SURI, Numerical approximation of the spectra of non-compact operators arising in buckling problems. J. Numer. Math. 10 (2002) 193-219. MR 1935966 (2003i:74046)
  • 10. E. B. DAVIES, Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1 (1998) 42-74. IMA J. Numer. Anal. (2004) 417-438. MR 1635727 (2000e:47043)
  • 11. E. B. DAVIES, M. PLUM, Spectral pollution. IMA J. Numer. Anal. 24 (2004) 417-438. MR 2068830 (2005c:47027b)
  • 12. T. KATO, On the upper and lower bounds of eigenvalues. J. Phys. Soc. Japan 4 (1949) 334-339. MR 0038738 (12:447b)
  • 13. T. KATO, Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Analyse Math. 6 (1958) 261-322. MR 0107819 (21:6541)
  • 14. T. KATO, Perturbation theory for linear operators, Springer-Verlag (1966). MR 0203473 (34:3324)
  • 15. M. LEVITIN, E. SHARGORODSKY, Spectral pollution and second order relative spectra for self-adjoint operators, IMA J. Numer. Anal. (2004), 393-416. MR 2068829 (2005c:47027a)
  • 16. M. MARLETTA, Neumann-Dirichlet maps and analysis of spectral pollution for non-self-adjoint elliptic PDEs with real essential spectrum. IMA J Numer Anal 30 (2010) 917-939. MR 2727810 (2011j:65257)
  • 17. E. SHARGORODSKY, Geometry of higher order relative spectra and projection methods. J. Operator Theory 44 (2000) 43-62. MR 1774693 (2001f:47004)
  • 18. J. RAPPAZ, J. SANCHEZ HUBERT, E. SANCHEZ PALENCIA, D. VASSILIEV, On spectral pollution in the finite element approximation of thin elastic membrane shells. Numer. Math. 75 (1997) 473-500. MR 1431212 (97k:65259)
  • 19. M. STRAUSS, Quadratic projection methods for approximating the spectrum of self-adjoint operators. IMA J Numer Anal 31 (2011) 40-60. MR 2755936 (2012e:65100)
  • 20. R. STRICHARTZ, The Way of Analysis. Jones and Bartlett (2000).
  • 21. S. ZIMMERMANN, U. MERTINS, Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwendungen 14 (1995) 327-345. MR 1337263 (96d:49046)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 47A75, 47B15

Retrieve articles in all journals with MSC (2010): 47A75, 47B15

Additional Information

Michael Strauss
Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, Wales, United Kingdom CF24 4AG

Keywords: Spectral pollution, second order relative spectrum, convergence to eigenvalues, convergence to eigenvectors, projection methods, finite-section method
Received by editor(s): May 10, 2011
Received by editor(s) in revised form: June 13, 2011, and February 28, 2012
Published electronically: April 9, 2013
Additional Notes: The author gratefully acknowledges the support of EPSRC grant no. EP/I00761X/1.
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society