Skip to main content
SpringerLink
Log in
Book cover

Frontiers in Numerical Analysis - Durham 2010 pp 1–77Cite as

Some Remarks on Eigenvalue Approximation by Finite Elements

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Computational Science and Engineering ((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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   59.95
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. A. Alonso. Error estimators for a mixed method. Numer. Math., 74(4):385–395, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. M. G. Armentano and C. Padra. A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math., 58(5):593–601, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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. I. Babuška and W. C. Rheinboldt. A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng., 12:1597–1615, 1978.

    Article  MATH  Google Scholar 

  14. I. Babuška and W. C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15(4):736–754, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  21. D. Boffi. Finite element approximation of eigenvalue problems. Acta Numer., 19:1–120, 2010.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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. 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. A. Bossavit. Solving Maxwell equations in a closed cavity and the question of spurious modes. IEEE Trans. on Magnetics, 26:702–705, 1990.

    Article  Google Scholar 

  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. 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. 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. C. Carstensen. A posteriori error estimate for the mixed finite element method. Math. Comp., 66(218):465–476, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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. 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. M. Costabel and M. Dauge. Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal., 151(3):221–276, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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. F. Gardini. A posteriori error estimates for eigenvalue problems in mixed form. PhD thesis, Università degli Studi di Pavia, 2005.

    Google Scholar 

  46. F. Gardini. Mixed approximation of eigenvalue problems: a superconvergence result. M2AN Math. Model. Numer. Anal., 43(5):853–865, 2009.

    Google Scholar 

  47. L. Gastaldi. Mixed finite element methods in fluid structure systems. Numer. Math., 74(2):153–176, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  48. S. Giani and I. G. Graham. A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal., 47(2):1067–1091, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  49. L. Grubišić and J. S. Ovall. On estimators for eigenvalue/eigenvector approximations. Math. Comp., 78(266):739–770, 2009.

    MATH  MathSciNet  Google Scholar 

  50. F. Hecht. FreeFem++, Third Edition, Version 3.12, 2011. http://www.freefem.org/ff++/.

  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. R. Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, 11:237–339, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  53. T. Kato. Perturbation theory for linear operators. Springer-Verlag, New York, 1976.

    Book  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. A. V. Knyazev. New estimates for Ritz vectors. Math. Comp., 66(219):985–995, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  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. W. G. Kolata. Approximation in variationally posed eigenvalue problems. Numer. Math., 29(2):159–171, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  59. I. Kossaczký. A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math., 55(3):275–288, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  63. P. Monk. Finite element methods for Maxwell’s equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.

    Book  MATH  Google Scholar 

  64. J. E. Pasciak and J. Zhao. Overlapping Schwarz methods in H(curl) on polyhedral domains. J. Numer. Math., 10(3):221–234, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  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. W. C. Rheinboldt. On a theory of mesh-refinement processes. SIAM J. Numer. Anal., 17(6):766–778, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  68. M.-C. Rivara. Design and data structure of fully adaptive, multigrid, finite-element software. ACM Trans. Math. Software, 10(3):242–264, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  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. R. Verfürth. A posteriori error estimates for nonlinear problems. Math. Comp., 62(206):445–475, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  71. R. Verfürth. A Review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner, 1996.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  74. J. Webb. Edge elements and what they can do for you. IEEE Trans. on Magnetics, 29:1460–1465, 1993.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata 1, I-27100, Pavia, Italy

    Daniele Boffi & Francesca Gardini

  2. Dipartimento di Matematica, Università di Brescia, via Valotti 9, I-25133, Brescia, Italy

    Lucia Gastaldi

Authors
  1. Daniele Boffi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesca Gardini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lucia Gastaldi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Boffi .

Editor information

Editors and Affiliations

  1. , Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, United Kingdom

    James Blowey

  2. , Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, United Kingdom

    Max Jensen

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Boffi, D., Gardini, F., Gastaldi, L. (2011). Some Remarks on Eigenvalue Approximation by Finite Elements. In: Blowey, J., Jensen, M. (eds) Frontiers in Numerical Analysis - Durham 2010. Lecture Notes in Computational Science and Engineering, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23914-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation