Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 4, pp 661–692 | Cite as

Computational Aspects of Pseudospectra in Hydrodynamic Stability Analysis

  • D. Gerecht
  • R. RannacherEmail author
  • W. Wollner


This paper addresses the analysis of spectrum and pseudospectrum of the linearized Navier–Stokes operator from the numerical point of view. The pseudospectrum plays a crucial role in linear hydrodynamic stability theory and is closely related to the non-normality of the underlying differential operator and the matrices resulting from its discretization. This concept offers an explanation for experimentally observed instability in situations when eigenvalue-based linear stability analysis would predict stability. Hence the reliable numerical computation of the pseudospectrum is of practical importance particularly in situations when the stationary “base flow” is not analytically but only computationally given. The proposed algorithm is based on a finite element discretization of the continuous eigenvalue problem and uses an Arnoldi-type method involving a multigrid component. Its performance is investigated theoretically as well as practically at several two-dimensional test examples such as the linearized Burgers equations and various problems governed by the Navier–Stokes equations for incompressible flow.


Navier–Stokes equations Linearized stability Pseudospectrum Finite element method Arnoldi method Non-normaloperators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Becker R., Braack M. et al.: A two-level stabilization scheme for the Navier-Stokes equations. In: Feistauer, M. (ed) Numerical Mathematics and Advanced Applications., pp. 123–130. Springer, Proc. ENUMATH 2003 (2004)CrossRefGoogle Scholar
  2. 2.
    Bramble J.H., Osborn J.E.: Rate of convergence estimates for nonselfadjoint eigenvalue approximations. Math. Comp. 27, 525–545 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chen H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44, 1–28 (2006)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ciarlet P.G.: Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam (1978)Google Scholar
  5. 5.
    Dunford N., Schwartz J.T.: Linear Operators. Part 1: General Theory. Interscience Publishers, New York (1957)Google Scholar
  6. 6.
    Duran R.G., Nochetto R.H.: Pointwise accuracy of a stable Petrov-Galerkin approximation to the Stokes problem. SIAM J. Numer. Anal. 6, 1395–1406 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Franca L.P., Frey S.L., Hughes T.J.R.: Stabilized finite element methods: II. The incompressible Navier–Stokes equations. Comput. Meth. Appl. Mech. Eng. 99, 209–233 (1992)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Linearized Steady problems, vol. 1. Nonlinear Steady Problems, vol. 2. Springer, Berlin, Heidelberg, New York (1998)Google Scholar
  9. 9.
    Gascoigne: High Performance Adaptive Finite Element Toolkit. University of Heidelberg
  10. 10.
    Gerecht, D.: Pseudospektren in der Hydrodynamische Stabilitätstheorie. Diploma thesis, Institute of Applied Mathematics, University of Heidelberg (2010)Google Scholar
  11. 11.
    Girault V., Nochetto R.H., Scott R.: Maximum-norm stability of the finite element Stokes projection. J. Math. Pures Appl. 84, 279–330 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Girault V., Raviart P.A.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin, Heidelberg (1986)zbMATHCrossRefGoogle Scholar
  13. 13.
    Heuveline V., Bertsch C.: On multigrid methods for the eigenvalue computation of non-selfadjoint elliptic operators. East-West J. Numer. Math. 8, 275–297 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Heuveline V., Rannacher R.: A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15, 1–32 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Heuveline, V., Rannacher, R.: Adaptive FE eigenvalue approximation with application to hydrodynamic stability analysis. In: Fitzgibbon, W., et al. (eds.) Proc. Int. Conf. Advances in Numerical Mathematics, Moscow, Sept. 16–17, 2005, pp. 109–140. Institute of Numerical Mathematics RAS, Moscow (2006)Google Scholar
  16. 16.
    Heywood J., Rannacher R., Turek S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Comput. Fluid Mech. 22, 325–352 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hughes T.J.R., Brooks A.N.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equation. Comput. Meth. Appl. Mech. Eng. 32, 199–259 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hughes T.J.R., Franca L.P., Balestra M.: A new finite element formulation for computational fluid dynamics: V. Circumvent the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation for the Stokes problem accommodating equal order interpolation. Comput. Meth. Appl. Mech. Eng. 59, 89–99 (1986)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Johnson C., Rannacher R., Boman M.: Numerics and hydrodynamic stability: towards error control in CFD. SIAM J. Numer. Anal. 32, 1058–1079 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin, Heidelberg, New York (1966)zbMATHGoogle Scholar
  21. 21.
    Kreiss H.O.: Über die Stabilitätsdefinition für Differenzengleichungen, die partielle Differentialgleichungen approximieren. Nordisk Tidskr. Informations-Behandling 2, 153–181 (1962)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Orszag S.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703 (1971)ADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Osborn J.E.: Spectral approximation for compact operators. Math. Comp. 29, 712–725 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Quarteroni A., Valli A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin, Heidelberg, New York (1991)Google Scholar
  25. 25.
    Rannacher R.: Finite element methods for the incompressible Navier–Stokes equations. In: Galdi, G.P., Heywood, J., Rannacher, R. (eds) Fundamental Directions in Mathematical Fluid Mechanics, pp. 191–293. Birkhäuser, Basel, Boston, Berlin (2000)CrossRefGoogle Scholar
  26. 26.
    Rannacher, R.: Incompressible viscous flow. In: Stein, E., et al. (eds.) Encyclopedia of Computational Mechanics, vol. 3 “Fluids”. Wiley, Chichester (2004)Google Scholar
  27. 27.
    Rannacher, R.: Adaptive FE eigenvalue approximation with application to hydrodynamic stability. In: Sequeira, A., Rannacher, R. (eds.) Proc. Int. Conf. “Mathematical Fluid Mechanics”, Estoril, Spain, May 21–25, 2007. Springer, Heidelberg (2009)Google Scholar
  28. 28.
    Rannacher R., Westenberger A., Wollner W.: Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error. J. Numer. Math. 18(4), 303–327 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company (1996)Google Scholar
  30. 30.
    Schäfer, M., Turek, S.: Benchmark computations of laminar flow around a cylinder. In: Hirschel, E.H. (ed.) Flow Simulation with High-Performance Computer II. Notes on Numerical Fluid Mechanics, vol. 52, pp. 547–566. Vieweg, Braunschweig, Wiesbaden (1996)Google Scholar
  31. 31.
    Scott L.R., Zhang S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Trefethen L.N.: Pseudospectra of linear operators. In: Kirchgässner, K., Mahrenholtz, G., Mennicken, R. (eds) ICIAM 95. Proc. Third Int. Congr. on Industrial and Applied Mathematics, pp. 401–434. Akademie Verlag, Berlin (1996)Google Scholar
  33. 33.
    Trefethen L.N.: Computation of pseudospectra. Acta Numer. 8, 247–295 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Trefethen L.N., Embree M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)zbMATHGoogle Scholar
  35. 35.
    Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: A new direction in hydrodynamical stability: beyond eigenvalues. Tech. Report CTC92TR115 12/92, Cornell Theory Center, Cornell University (1992)Google Scholar
  36. 36.
    Trefethen L.N., Reddy S.C., Driscoll T.A.: Hydrodynamic stability without eigenvalues. Science (New Series) 261(5121), 578–584 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Westenberger, A.: Numerische Lösung von Eigenwertaufgaben unsymmetrischer partieller Differentialoperatoren. Diploma thesis, Institute of Applied Mathematics, University of Heidelberg (2009)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany
  2. 2.Department of MathematicsUniversity of HamburgHamburgGermany

Personalised recommendations