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Some Remarks on Eigenvalue Approximation by Finite Elements

  • Daniele Boffi
  • Francesca Gardini
  • Lucia Gastaldi
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 85)

Abstract

The aim of this paper is to supplement the results of Boffi (Acta Numer. 19:1–120, 2010) with some additional remarks. In particular we deal with three distinct topics: we review some tutorial examples in one dimension and provide numerical codes for them; we analyze the case of multiple eigenvalues and show some numerical; we review a posteriori error analysis for eigenvalue problems.

Keywords

Eigenvalue Problem Posteriori Error Posteriori Error Estimate Error Indicator Multiple Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Ainsworth and J. T. Oden. A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2000.Google Scholar
  2. 2.
    A. Alonso. Error estimators for a mixed method. Numer. Math., 74(4):385–395, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Alonso, A. Dello Russo, C. Otero-Souto, C. Padra, and R. Rodríguez. An adaptive finite element scheme to solve fluid-structure vibration problems on non-matching grids. Comput. Vis. Sci., 4(2):67–78, 2001. Second AMIF International Conference (Il Ciocco, 2000).Google Scholar
  4. 4.
    A. Alonso, A. Dello Russo, C. Padra, and R. Rodríguez. 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):25–59 (2002), 2001. A posteriori error estimation and adaptive computational methods.Google Scholar
  5. 5.
    A. Alonso, A. Dello Russo, and V. Vampa. A posteriori error estimates in finite element solution of structure vibration problems with applications to acoustical fluid-structure analysis. Comput. Mech., 23(3):231–239, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Alonso, A. Dello Russo, and V. Vampa. A posteriori error estimates in finite element acoustic analysis. J. Comput. Appl. Math., 117(2):105–119, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potential in three-dimensional nonsmooth domains. Math Methods Appl. Sci., 21(9):823–864, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    P. M. Anselone. Collectively compact operator approximation theory and applications to integral equations. Prentice-Hall Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis, Prentice-Hall Series in Automatic Computation.Google Scholar
  9. 9.
    M. G. Armentano and C. Padra. A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math., 58(5):593–601, 2008.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.), 47(2):281–354, 2010.Google Scholar
  11. 11.
    I. Babuška and R. Narasimhan. The Babuška-Brezzi condition and the patch test: an example. Comput. Methods Appl. Mech. Engrg., 140(1-2):183–199, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    I. Babuška and J. Osborn. Eigenvalue problems. In Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pages 641–787. North-Holland, Amsterdam, 1991.Google Scholar
  13. 13.
    I. Babuška and W. C. Rheinboldt. A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng., 12:1597–1615, 1978.zbMATHCrossRefGoogle Scholar
  14. 14.
    I. Babuška and W. C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15(4):736–754, 1978.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    R. E. Bank. The efficient implementation of local mesh refinement algorithms. In Adaptive computational methods for partial differential equations (College Park, Md., 1983), pages 74–81. SIAM, Philadelphia, PA, 1983.Google Scholar
  16. 16.
    R. E. Bank. PLTMG: a software package for solving elliptic partial differential equations. Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Users’ guide 8.0.Google Scholar
  17. 17.
    R. E. Bank, A. H. Sherman, and A. Weiser. Refinement algorithms and data structures for regular local mesh refinement. In Scientific computing (Montreal, Que., 1982), IMACS Trans. Sci. Comput., I, pages 3–17. IMACS, New Brunswick, NJ, 1983.Google Scholar
  18. 18.
    K.-J. Bathe, C. Nitikitpaiboon, and X. Wang. A mixed displacement-based finite element formulation for acoustic fluid-structure interaction. Comput. & Structures, 56(2-3):225–237, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Bermúdez, R. Durán, M. A. Muschietti, R. Rodríguez, and J. Solomin. Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal., 32(4):1280–1295, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Bermúdez and D. G. Pedreira. Mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides. Numer. Math., 61(1):39–57, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Boffi. Finite element approximation of eigenvalue problems. Acta Numer., 19:1–120, 2010.MathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Boffi, F. Brezzi, and L. Gastaldi. On the convergence of eigenvalues for mixed formulations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25(1-2):131–154 (1998), 1997. Dedicated to Ennio De Giorgi.Google Scholar
  23. 23.
    D. Boffi, F. Brezzi, and L. Gastaldi. On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp., 69(229):121–140, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    D. Boffi, C. Chinosi, and L. Gastaldi. Approximation of the grad div operator in nonconvex domains. CMES Comput. Model. Eng. Sci., 1(2):31–43, 2000.MathSciNetGoogle Scholar
  25. 25.
    D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia. Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal., 36(4):1264–1290 (electronic), 1999.Google Scholar
  26. 26.
    D. Boffi and C. Lovadina. Remarks on augmented Lagrangian formulations for mixed finite element schemes. Boll. Un. Mat. Ital. A (7), 11(1):41–55, 1997.Google Scholar
  27. 27.
    A. Bossavit. Solving Maxwell equations in a closed cavity and the question of spurious modes. IEEE Trans. on Magnetics, 26:702–705, 1990.CrossRefGoogle Scholar
  28. 28.
    J. H. Bramble, J. E. Pasciak, and A. V. Knyazev. A subspace preconditioning algorithm for eigenvector/eigenvalue computation. Adv. Comput. Math., 6(2):159–189 (1997), 1996.Google Scholar
  29. 29.
    F. Brezzi and M. Fortin. Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.Google Scholar
  30. 30.
    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods. Scientific Computation. Springer-Verlag, Berlin, 2006. Fundamentals in single domains.Google Scholar
  31. 31.
    C. Carstensen. A posteriori error estimate for the mixed finite element method. Math. Comp., 66(218):465–476, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    C. Carstensen and J. Gedicke. An oscillation-free adaptive FEM for symmetric eigenvalue problems. Technical report, DFG Research Center MATHEON, Berlin, 2008.Google Scholar
  33. 33.
    C. Carstensen and R. Verfürth. Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal., 36(5):1571–1587, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    H. Chen and R. Taylor. Vibration analysis of fluid-solid systems using a finite element displacement formulation. Int. J. Numer. Methods Eng., 29:683–698, 1990.zbMATHCrossRefGoogle Scholar
  35. 35.
    P. G. Ciarlet. The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4.Google Scholar
  36. 36.
    P. Clément. Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9(R-2):77–84, 1975.Google Scholar
  37. 37.
    M. Costabel and M. Dauge. Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal., 151(3):221–276, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    X. Dai, J. Xu, and A. Zhou. Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math., 110(3):313–355, 2008.zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    M. Dauge. Benchmark computations for Maxwell equations for the approximation of highly singular solutions. http://perso.univ-rennes1.fr/monique.dauge/benchmax.html.
  40. 40.
    R. Durán, L. Gastaldi, and C. Padra. A posteriori error estimators for mixed approximations of eigenvalue problems. Math. Models Methods Appl. Sci., 9(8):1165–1178, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    R. Durán, C. Padra, and R. Rodríguez. A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Models Methods Appl. Sci., 13(8):1219–1229, 2003.zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    E M. Garau and P. Morin. Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems. Cuad. Mat. Mec., page 30, 2009.Google Scholar
  43. 43.
    E. M. Garau, P. Morin, and C. Zuppa. Convergence of adaptive finite element methods for eigenvalue problems. Math. Models Methods Appl. Sci., 19(5):721–747, 2009.zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    F. Gardini. A posteriori error estimates for an eigenvalue problem arising from fluid-structure interaction. Istit. Lombardo Accad. Sci. Lett. Rend. A, 138:17–34 (2005), 2004.Google Scholar
  45. 45.
    F. Gardini. A posteriori error estimates for eigenvalue problems in mixed form. PhD thesis, Università degli Studi di Pavia, 2005.Google Scholar
  46. 46.
    F. Gardini. Mixed approximation of eigenvalue problems: a superconvergence result. M2AN Math. Model. Numer. Anal., 43(5):853–865, 2009.Google Scholar
  47. 47.
    L. Gastaldi. Mixed finite element methods in fluid structure systems. Numer. Math., 74(2):153–176, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    S. Giani and I. G. Graham. A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal., 47(2):1067–1091, 2009.zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    L. Grubišić and J. S. Ovall. On estimators for eigenvalue/eigenvector approximations. Math. Comp., 78(266):739–770, 2009.zbMATHMathSciNetGoogle Scholar
  50. 50.
    F. Hecht. FreeFem++, Third Edition, Version 3.12, 2011. http://www.freefem.org/ff++/.
  51. 51.
    V. Heuveline and R. Rannacher. A posteriori error control for finite approximations of elliptic eigenvalue problems. Adv. Comput. Math., 15(1-4):107–138 (2002), 2001.Google Scholar
  52. 52.
    R. Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, 11:237–339, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    T. Kato. Perturbation theory for linear operators. Springer-Verlag, New York, 1976.zbMATHCrossRefGoogle Scholar
  54. 54.
    F. Kikuchi. Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Comput. Methods Appl. Mech. Engrg., 64:509–521, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    A. V. Knyazev. Sharp a priori error estimates of the rayleigh-ritz method without assumptions of fixed sign or compactness. Math. Notes, 38:998–1002, 1986.Google Scholar
  56. 56.
    A. V. Knyazev. New estimates for Ritz vectors. Math. Comp., 66(219):985–995, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    A. V. Knyazev and J. E. Osborn. New a priori FEM error estimates for eigenvalues. SIAM J. Numer. Anal., 43(6):2647–2667 (electronic), 2006.Google Scholar
  58. 58.
    W. G. Kolata. Approximation in variationally posed eigenvalue problems. Numer. Math., 29(2):159–171, 1978.zbMATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    I. Kossaczký. A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math., 55(3):275–288, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    M. G. Larson. 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 (electronic), 2000.Google Scholar
  61. 61.
    C. Lovadina, M. Lyly, and R. Stenberg. A posteriori estimates for the Stokes eigenvalue problem. Numer. Methods Partial Differential Equations, 25(1):244–257, 2009.zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    D. Mao, L. Shen, and A. Zhou. Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates. Adv. Comput. Math., 25(1-3):135–160, 2006.zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    P. Monk. Finite element methods for Maxwell’s equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.zbMATHCrossRefGoogle Scholar
  64. 64.
    J. E. Pasciak and J. Zhao. Overlapping Schwarz methods in H(curl) on polyhedral domains. J. Numer. Math., 10(3):221–234, 2002.zbMATHMathSciNetCrossRefGoogle Scholar
  65. 65.
    P.-A. Raviart and J.-M. Thomas. A mixed finite element method for second order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical Aspects of the Finite Element Method, volume 606 of Lecture Notes in Math., pages 292–315, New York, 1977. Springer-Verlag.Google Scholar
  66. 66.
    W. C. Rheinboldt. On a theory of mesh-refinement processes. SIAM J. Numer. Anal., 17(6):766–778, 1980.zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    M.-C. Rivara. Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. Internat. J. Numer. Methods Engrg., 20(4):745–756, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    M.-C. Rivara. Design and data structure of fully adaptive, multigrid, finite-element software. ACM Trans. Math. Software, 10(3):242–264, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    G. Valle. Problemi agli autovalori per equazioni alle derivate parziali: autovalori multipli. Tesi di laurea, Università degli Studi di Pavia, 2011.Google Scholar
  70. 70.
    R. Verfürth. A posteriori error estimates for nonlinear problems. Math. Comp., 62(206):445–475, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    R. Verfürth. A Review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner, 1996.Google Scholar
  72. 72.
    T. F. Walsh, G. M. Reese, and U. L. Hetmaniuk. Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures. Comput. Methods Appl. Mech. Engrg., 196(37-40):3614–3623, 2007.zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    X. Wang and K.-J. Bathe. On mixed elements for acoustic fluid-structure interactions. Math. Models Methods Appl. Sci., 7(3):329–343, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    J. Webb. Edge elements and what they can do for you. IEEE Trans. on Magnetics, 29:1460–1465, 1993.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniele Boffi
    • 1
  • Francesca Gardini
    • 1
  • Lucia Gastaldi
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly
  2. 2.Dipartimento di MatematicaUniversità di BresciaBresciaItaly

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