Numerische Mathematik

, Volume 119, Issue 3, pp 557–583 | Cite as

An adaptive homotopy approach for non-selfadjoint eigenvalue problems

  • C. Carstensen
  • J. Gedicke
  • V. Mehrmann
  • A. Miedlar
Article

Abstract

This paper presents adaptive algorithms for eigenvalue problems associated with non-selfadjoint partial differential operators. The basis for the developed algorithms is a homotopy method which departs from a well-understood selfadjoint problem. Apart from the adaptive grid refinement, the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources. The first algorithm balances the homotopy, discretization and approximation errors with respect to a fixed stepsize τ in the homotopy. The second algorithm combines the adaptive stepsize control for the homotopy with an adaptation in space that ensures an error below a fixed tolerance ε. The outcome of the analysis leads to the third algorithm which allows the complete adaptivity in space, homotopy stepsize as well as the iterative algebraic eigenvalue solver. All three algorithms are compared in numerical examples.

Mathematics Subject Classification (2010)

65F15 65N15 65N25 65N50 65M60 65H20 

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References

  1. 1.
    Ainsworth M., Oden J.T.: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, Inc., New Jersey (2000)MATHCrossRefGoogle Scholar
  2. 2.
    Aitken A.C.: On Bernoulli’s numerical solution of algebraic equations. Proc. Royal Soc. Edinb. 46, 289–305 (1926)MATHGoogle Scholar
  3. 3.
    Alonso A., Dello Russo A., Padra C., Rodríguez R.: A posteriori error estimates and a local refinement strategy for a finite element method to solve structural-acoustic vibration problems. Adv. Comput. Math. 15(1–4), 25– (2001)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Argentati M.E., Knyazev A.V., Paige C.C., Panayotov I.: Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix. SIAM J. Matrix Anal. Appl. 30(2), 548–559 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Babuška I., Osborn J.E.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp. 52(186), 275–297 (1989)MathSciNetMATHGoogle Scholar
  6. 6.
    Babuška, I., Osborn J.E.: Eigenvalue Problems. Handbook of Numerical Analysis, vol. 2 (1991)Google Scholar
  7. 7.
    Bai Z., Demmel J., Dongarra J., Ruhe A., van der Vorst H.: Templates for the Solution of Algebraic Eigenvalue Problem. A Practical Guide. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  8. 8.
    Bangerth W., Rannacher R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)MATHGoogle Scholar
  9. 9.
    Boffi D., Fernandes P., Gastaldi L., Perugia I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36(4), 1264–1290 (1999)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Braack M., Ern A.: A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1(2), 221–238 (2003)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods Texts in Applied Mathematics. 2nd edn. Springer, Berlin (2002)Google Scholar
  12. 12.
    Cartensen, C., Gedicke, J.: An oscillation-free adaptive FEM for symmetric eigenvalue problems. Preprint 489, DFG Research Center Matheon, Straße des 17. Juni 136, D-10623 Berlin (2008)Google Scholar
  13. 13.
    Chatelin F.: Spectral Approximation of Linear Operators. Academic Press, New York (1983)MATHGoogle Scholar
  14. 14.
    Dörfler W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Durán R.G., Padra C., Rodriguez R.: A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 13, 1219–1229 (2003)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Evans L.C.: Partial differential equations. American Mathematical Society, Providence (2000)Google Scholar
  17. 17.
    Garau E.M., Morin P., Zuppa C.: Convergence of adaptive finite element methods for eigenvalue problems. Math. Models Methods Appl. Sci. 19(5), 721–747 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Gedicke J., Carstensen C.: A posteriori error estimators for non-symmetric eigenvalue problems. Preprint 659, DFG Research Center Matheon, Str. des 17. Juni 136, D-10623 Berlin, (2009)Google Scholar
  19. 19.
    Giani S., Graham I.G.: A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 47(2), 1067–1091 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Golub G.H., Van Loan C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  21. 21.
    Grubišić L., Ovall J.S.: On estimators for eigenvalue/eigenvector approximations. Math. Comp. 78(266), 739–770 (2009)MathSciNetMATHGoogle Scholar
  22. 22.
    Hairer E., Nørsett S.P., Wanner G.: Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd edn. Springer, Berlin (1993)MATHGoogle Scholar
  23. 23.
    Heiserer, D., Zimmer, H., Schäfer, M., Holzheuer C., Kondziella R.: Formoptimierung in der frühen Phase der Karosserieentwicklung. Vdi-Berichte 1846, Würzburg (2004)Google Scholar
  24. 24.
    Hetmaniuk U.L., Lehoucq R.B.: Uniform accuracy of eigenpairs from a shift-invert lanczos method. SIAM J. Matrix Anal. Appl. 28, 927–948 (2006)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Heuveline V., Rannacher R.: A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comp. Math. 15, 107–138 (2001)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Kato T.: Pertubation Theory for linear Operators. Springer, Berlin (1980)Google Scholar
  27. 27.
    Knyazev A.V.: New estimates for Ritz vectors. Math. Comp. 66(219), 985–995 (1997)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Knyazev A.V., Argentati M.E.: Rayleigh-Ritz majorization error bounds with applications to fem. SIAM. J. Matrix Anal. Appl. 31(3), 1521–1537 (2010)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Knyazev A.V., Osborn J.E.: New a priori FEM error estimates for eigenvalues. SIAM J. Numer. Anal. 43(6), 2647–2667 (2006)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Larson M.G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38(2), 608–625 (2000)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Larsson S., Thomée V.: Partial Differential Equations with Numerical Methods. Springer, Berlin (2003)MATHGoogle Scholar
  32. 32.
    Lehoucq R.B., Sorensen D.C., Yang C.: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia (1998)Google Scholar
  33. 33.
    Li T.Y., Zeng Z.: Homotopy-determinant algorithm for solving non-symmetric eigenvalue problems. Math. Comp. 59, 483–502 (1992)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Li T.Y., Zeng Z.: The homotopy continuation algorithm for the real nonsymmetric eigenproblem: Further development and implementation. SIAM J. Sci. Comp. 20, 1627–1651 (1999)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Li T.Y., Zeng Z., Cong L.: Solving eigenvalue problems of real nonsymmetrric matrices with real homotopies. SIAM J. Numer. Anal. 29, 229–248 (1992)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Lui S.H., Golub G.H.: Homotopy method for the numerical solution of the eigenvalue problem of slef-adjoint partial differential operators. Numer. Algorithms 10, 363–378 (1995)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Lui S.H., Keller H.B., Kwok T.W.C.: Homotopy method for the large sparse real nonsymmetric eigenvalue problem. SIAM J. Matrix Anal. Appl. 18, 312–333 (1997)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Mao D., Shen L., Zhou A.: Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates. Adv. Comp. Math. 25, 135–160 (2006)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    MATLAB, Version 7.10.0.499 (R2010a). The MathWorks, inc., 24 Prime Park Way, Natick, MA 01760-1500, USA (2010)Google Scholar
  40. 40.
    Mehrmann V., Miedlar A.: Adaptive computation of smallest eigenvalues of elliptic partial differential equations. Numer. Linear Algebra Appl. 18, 387–409 (2011)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Neymeyr K.: A posteriori error estimation for elliptic eigenproblems. Numer. Linear Algebra Appl. 9(4), 263–279 (2002)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Parlett B.N.: The symmetric eigenvalue problem. SIAM, Philadelphia (1998)MATHGoogle Scholar
  43. 43.
    Raviart P.A., Thomas J.M.: Introduction á l’Analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983)MATHGoogle Scholar
  44. 44.
    Saad Y.: Numerical methods for large eigenvalue problems. Manchester University Press, Manchester (1992)MATHGoogle Scholar
  45. 45.
    Sauter S.: hp-finite elements for elliptic eigenvalue problems: error estimates which are explicit with respect to λ, h, and p. SIAM J. Numer. Anal. 48(1), 95–108 (2010)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Stewart G.W., Sun J.G.: Matrix perturbation theory. Academic Press Inc., Boston (1990)MATHGoogle Scholar
  47. 47.
    Strang G., Fix G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)MATHGoogle Scholar
  48. 48.
    Trefethen L.N., Betcke T.: Computed eigenmodes of planar regions. Contemp. Math. 412, 297–314 (2006)MathSciNetGoogle Scholar
  49. 49.
    Verfürth R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley and Teubner, San Francisco (1996)MATHGoogle Scholar
  50. 50.
    Xu J., Zhou A.: A two-grid discretization scheme for eigenvalue problems. Math. Comp. 70(233), 17–25 (2001)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • C. Carstensen
    • 1
    • 2
  • J. Gedicke
    • 1
  • V. Mehrmann
    • 3
  • A. Miedlar
    • 3
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea
  3. 3.Institut für MathematikBerlinGermany

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