BIT Numerical Mathematics

, Volume 52, Issue 4, pp 801–825 | Cite as

Eigenvalue enclosures and convergence for the linearized MHD operator

  • Lyonell Boulton
  • Michael Strauss


Computation of certified enclosures for eigenvalues of benchmark linear magnetohydrodynamics (MHD) operators in the plane slab and cylindrical pinch configuration is discussed. For the plane slab, the proposed method relies upon the formulation of an eigenvalue problem associated to the Schur complement. This leads to highly accurate upper bounds for the eigenvalue. For the cylindrical configuration, a direct application of this formulation is possible, however, it cannot be rigourously justified. Therefore in this case the proposed method relies on a specialized technique based on a method proposed by Zimmermann and Mertins. It turns out that this technique is also applicable for finding accurate complementary bounds in the case of the plane slab. Convergence rates for both approaches are established.


Eigenvalue enclosures Magnetohydrodynamics Schur complement Spectral pollution 

Mathematics Subject Classification (2010)

65F15 15A18 47B36 47B39 81-08 


  1. 1.
    Atkinson, F., Langer, H., Mennicken, R., Shkalikov, A.: The essential spectrum of some matrix operators. Math. Nachr. 167, 5–20 (1994) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Behnke, H., Mertins, U.: Bounds for eigenvalues with the use of finite elements. In: Perspectives on Enclosure Methods, Karlsruhe, 2000, pp. 119–131. Springer, Berlin (2001) CrossRefGoogle Scholar
  3. 3.
    Behnke, H.: Lower and upper bounds for sloshing frequencies. Int. Ser. Numer. Math. 157, 13–22 (2009) MathSciNetGoogle Scholar
  4. 4.
    Boulton, L., Strauss, M.: On the convergence of second-order spectra and multiplicity. Proc. R. Soc. A 467, 264–275 (2011) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Davies, E.B.: Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1, 42–74 (1998) MathSciNetMATHGoogle Scholar
  6. 6.
    Chatelin, F.: Spectral Approximation of Linear Operators. Academic Press, San Diego (1983) MATHGoogle Scholar
  7. 7.
    Davies, E.B., Plum, M.: Spectral pollution. IMA J. Numer. Anal. 24, 417–438 (2004) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Eschwe, D., Langer, M.: Variational principles for eigenvalues of self-adjoint operator functions. Inter. Equ. Oper. Theory 49, 287–321 (2004) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Goerisch, F., Haunhorst, H.: Eigenwertschranken fur Eigenwertaufgaben mit Partiellen Differentialgleinschungen. Z. Angew. Math. Mech. 65, 129–135 (1985) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kako, T.: Essential spectrum of linearized MHD operator in cylindrical region. Z. Angew. Math. Phys. 38, 433–449 (1987) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kako, T., Descloux, J.: Spectral approximation for the linearized MHD operator in cylindrical region. Jpn. J. Ind. Appl. Math. 8, 221–244 (1991) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kato, T.: On the upper and lower bounds of eigenvalues. J. Phys. Soc. Jpn. 4, 334–339 (1949) CrossRefGoogle Scholar
  13. 13.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966) MATHGoogle Scholar
  14. 14.
    Levitin, M., Shargorodsky, E.: Spectral pollution and second order relative spectra for self-adjoint operators. IMA J. Numer. Anal. 24, 393–416 (2004) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kraus, M., Langer, M., Tretter, C.: Variational principles and eigenvalue estimates for unbounded block operator matrices and applications. J. Comput. Appl. Math. 171, 311–334 (2004) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lehmann, N.J.: Optimale Eigenwerteinschliessungen. Numer. Math. 5, 246–272 (1963) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Lifschitz, A.: Magnetohydrodynamics and Spectral Theory. Kluwer Academic, Norwell (1989) MATHCrossRefGoogle Scholar
  18. 18.
    Raikov, G.: The spectrum of a linear magnetohydrodynamic model with cylindrical symmetry. C. R. Acad. Bulgare Sci. 39, 17–20 (1986) MathSciNetGoogle Scholar
  19. 19.
    Raikov, G.: The spectrum of a linear magnetohydrodynamic model with cylindrical symmetry. Arch. Ration. Mech. Anal. 116, 161–198 (1991) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Rappaz, J.: Approximation of the spectrum of a non-compact operator given by the magnetohydrodynamic stability of a plasma. Numer. Math. 28, 15–24 (1977) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Shargorodsky, E.: Geometry of higher order relative spectra and projection methods. J. Oper. Theory 44, 43–62 (2000) MathSciNetMATHGoogle Scholar
  22. 22.
    Strauss, M.: Quadratic projection methods for approximating the spectrum of self-adjoint operators IMA. J. Numer. Anal. 31, 40–60 (2011) MATHCrossRefGoogle Scholar
  23. 23.
    Strauss, M.: The second order spectrum and optimal convergence. Preprint (2012) Google Scholar
  24. 24.
    Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2007) Google Scholar
  25. 25.
    Zimmermann, S., Mertins, U.: Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwend. 14, 327–345 (1995) MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghUK

Personalised recommendations