Skip to main content
SpringerLink
Log in
Book cover

Forging Connections between Computational Mathematics and Computational Geometry pp 43–52Cite as

A Least Squares Approach for Exponential Rate of Convergence of Eigenfunctions of Second-Order Elliptic Eigenvalue Problem

  • Conference paper

  • 805 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 124))

Abstract

In this paper, we show the convergence estimates for eigenvalues and eigenvectors of second-order elliptic eigenvalue problems using spectral element method. A least squares approach is used to prove that the eigenvalues and eigenvectors converge exponentially in P, degree of polynomials, when the boundary of the domains is to be assumed sufficiently smooth and the coefficients of the differential operator are analytic.

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   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
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

References

  1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Babuska, I., Guo, B.Q.: The h-p version of the finite element method on domains with curved boundaries. SIAM J. Numer. Anal. 25(4), 837–861 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuska, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Finite element methods (part II). Handbook of numerical analysis, vol. 2. (1991)

    Google Scholar 

  4. Balyan, L.K., Dutt, P., Rathore, R.K.S.: Least squares h-p spectral element method for elliptic eigenvalue problems. Appl. Math. Comput. 218(19), 9596–9613 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Bramble, J.H., Osborn, J.E.: Rate of convergence estimates for non-selfadjoint eigenvalue approximation. Math. Comput. 27, 525–549 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramble, J.H., Schatz, A.H.: Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions. Commun. Pure Appl. Math. 23, 653–675 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dutt, P., Kishore, N., Upadhyay, C.S.: Nonconforming h-p spectral element methods for elliptic problems. Proc. Indian Acad. Sci. (Math. Sci.), Springer 117(1), 109–145 (2007)

    Google Scholar 

  8. Kato, T.: Perturbation theory for linear operators, 2nd edn. Springer, Berlin/New York (1980)

    MATH  Google Scholar 

  9. Osborn, J.E.: Spectral approximation for compact operators. Math. Comput. 29, 712–725 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Schwab, Ch.: p and h-p Finite Element Methods. Clarendon Press, Oxford (1998)

    Google Scholar 

  11. Tomar, S.K.: h-p spectral element methods for elliptic problems on non-smooth domains using parallel computers, Ph.D. Thesis (IIT Kanpur, INDIA), 2001; Reprint available as Tec. Rep. no. 1631, Department of Applied Mathematics, University of Twente, The Netherlands. http://www.math.utwente.nl/publications

Download references

Author information

Authors and Affiliations

  1. IIIDM-Jabalpur, Jabalpur, MP, 482005, India

    Lokendra K. Balyan

Authors
  1. Lokendra K. Balyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lokendra K. Balyan .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Global Science and Technology Forum, Singapore, Singapore

    Anton Ravindran (Managing Editor, President) (Managing Editor, President)

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Balyan, L.K. (2016). A Least Squares Approach for Exponential Rate of Convergence of Eigenfunctions of Second-Order Elliptic Eigenvalue Problem. In: Chen, K., Ravindran, A. (eds) Forging Connections between Computational Mathematics and Computational Geometry. Springer Proceedings in Mathematics & Statistics, vol 124. Springer, Cham. https://doi.org/10.5176/2251-1911_CMCGS14.32_5

Download citation

Publish with us

Policies and ethics

Search

Navigation