Skip to main content
SpringerLink
Log in

Supercloseness Between the Elliptic Projection and the Approximate Eigenfunction and Its Application to a Postprocessing of Finite Element Eigenvalue Problems

  • Conference paper
Numerical Analysis and Its Applications (NAA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3401))

Included in the following conference series:

Abstract

An estimate confirming the supercloseness between the Ritz projection and the corresponding eigenvectors, obtained by finite element method, is hereby proved. This result is true for a large class of self-adjoint 2m–order elliptic operators. An application of this theorem to the superconvergence postprocessing patch-recovery technique for finite element eigenvalue problems is also presented. Finally, the theoretical investigations are supported by numerical experiments.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  2. Babuska, I., Osborn, J.: Eigenvalue Problems. Handbook of Numer. Anal., vol. II. North-Holland, Amsterdam (1991)

    Google Scholar 

  3. Pierce, J.G., VCarga, R.S.: Higher order convergence result for the Rayleigh-Ritz method applied to eigenvalue problems: Improved error bounds for eigenfunctions. Numer. Math. 19, 155–169 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  4. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  5. Wahlbin, L.B.: Superconvergence in Galerkin FEM. Springer, Berlin (1995)

    Google Scholar 

  6. Lin, Q., Yan, N., Zhou, A.: A rectangular test for interpolated finite elements. In: Proceedings of Systems Science & Systems Engineering, pp. 217–229. Culture Publish Co. (1991)

    Google Scholar 

  7. Andreev, A.B.: Superconvergence of the gradient of finite element eigenfunctions. C.R. Acad. Bulg. Sci. 43, 9–11 (1990)

    MATH  Google Scholar 

  8. Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  9. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Engrg. 33, 1331–1364 (1992)

    MATH  MathSciNet  Google Scholar 

  10. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley Interscience, New York (2000)

    MATH  Google Scholar 

  11. Zhang, Z., Liu, R.: Ultraconcergence of ZZ patch recovery at mesh symmetry points. Numer. Math. 95, 781–801 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Andreev, A.B., Dimov, T.T., Racheva, M.R.: One-dimensional patch-recovery finite element method for fourth-order elliptic problems (in this issue)

    Google Scholar 

  13. Racheva, M.R., Andreev, A.B.: Variational aspects of one-dimensional fourth-order problems with eigenvalue parameter in the boundary conditions. Sib. J.N.M. 4(5), 373–380 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technical University of Gabrovo & IPP – Bulgarian Academy of Sciencies,  

    Andrey B. Andreev

Authors
  1. Andrey B. Andreev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Kowloon, P.O. Box, Hong Kong

    Zhilin Li

  2. Department of Mathematics, University of Rousse, 7017, Rousse, Bulgaria

    Lubin Vulkov

  3. Informatics & Mathematical Modeling, Technical University of Denmark, DK-2800, Lyngby, Denmark

    Jerzy Waśniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Andreev, A.B. (2005). Supercloseness Between the Elliptic Projection and the Approximate Eigenfunction and Its Application to a Postprocessing of Finite Element Eigenvalue Problems. In: Li, Z., Vulkov, L., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2004. Lecture Notes in Computer Science, vol 3401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31852-1_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-31852-1_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24937-5

  • Online ISBN: 978-3-540-31852-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation