Skip to main content

Boundary collocation in Fejér points for computing eigenvalues and eigenfunctions of the Laplacian

Part of the Lecture Notes in Mathematics book series (LNM,volume 1287)

Abstract

The boundary collocation method is applied to the computation of eigenvalues and eigenfunctions of the Laplace operator on planar simply connected regions with smooth boundaries. For convex regions we seek to approximate the eigenfunctions by a linear combination of basis functions that contain Bessel functions of the first kind. Our method differs from related schemes proposed previously in that we distribute the collocation points differently, and we use a different iterative scheme for computing eigenvalues and eigenfunctions. This makes our method both faster and more accurate. For nonconvex regions rapid convergence generally can be achieved only if the eigenfunctions are approximated by functions with singular points in the finite plane. A boundary collocation method with such basis functions is also described.

Keywords

  • Error Bound
  • Iterative Scheme
  • Collocation Point
  • Parametric Representation
  • Lower 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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R.H.T. Bates and F.L. Ng, Point matching computation of transverse resonances, Int. J. Numer. Meth. Engng., 6 (1973), 155–168.

    CrossRef  MATH  Google Scholar 

  2. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Wiley, New York, 1953.

    Google Scholar 

  3. J.H. Curtiss, Transfinite diameter and harmonic polynomial interpolation, J. d’Analyse Math., 22 (1969), 371–389.

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. S.C. Eisenstat, On the rate of convergence of the Bergman-Vekua method for the numerical solution of elliptic boundary value problems, SIAM J. Numer. Anal., 11 (1974), 654–680.

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. L. Fox, P. Henrici and C. Moler, Approximation and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal, 4 (1967), 89–102.

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. B.E. Fischer, Approximationssätze für Lösungen der Helmholtzgleichung und ihre Anwendung auf die Berechnung von Eigenwerten, Ph.D. thesis, ETH, Zürich, 1983.

    Google Scholar 

  7. D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel, 1980.

    CrossRef  MATH  Google Scholar 

  8. M.H. Gutknecht, Numerical conformal mapping methods based on function conjugation, J. Comput. Appl. Math., 14 (1986), 31–77, in [T].

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. P. Henrici, Applied and Computational Complex Analysis, vol. 3, Wiley, New York, 1986.

    Google Scholar 

  10. R. Hettich and P. Zenke, Two case-studies in parametric semi-infinite programming, in Systems and Optimization, eds. A. Bagchi and H. Th. Jongen, Lecture Notes in Control and Information Sciences, No. 66, Springer, Berlin, 1985, 132–155.

    CrossRef  Google Scholar 

  11. J.R. Kuttler and V.G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26 (1984), 163–193.

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. C.B. Moler, Accurate bounds for the eigenvalues of the Laplacian and applications to rhombical domains, Report # CS 121, Computer Science Department, Stanford University, 1969.

    Google Scholar 

  13. C.B. Moler and L.E. Payne, Bounds for eigenvalues and eigenvectors of symmetric operators, SIAM J. Numer. Anal., 5 (1968), 64–70.

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. L. Reichel, On the computation of eigenvalues of the Laplacian by the boundary collocation method, in Approximation Theory V, eds. C.K. Chui et al., Academic Press, Boston, 1986, pp. 539–543.

    Google Scholar 

  15. L. Reichel, On complex rational approximation by interpolation at preselected nodes, Complex Variables: Theory and Appl., 4 (1984), 63–87.

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. L. Reichel, On the determination of boundary collocation points for solving some problems for the Laplace operator, J. Comput. Appl. Math., 11 (1984), 173–196.

    CrossRef  MathSciNet  Google Scholar 

  17. L. Reichel, Numerical methods for analytic continuation and mesh generation, Constr. Approx., 2 (1986), 23–39.

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. B.E. Spielman and R.F. Harrington, Waveguides of arbitrary cross section by solution of a nonlinear integral eigenvalue equation, IEEE Trans. Microwave Theory Techn., MTT-20 (1972), 578–585.

    CrossRef  Google Scholar 

  19. L.N. Trefethen, ed., Numerical Conformal Mapping, J. Comput. Appl. Math., 14 (1986).

    Google Scholar 

  20. I.N. Vekua, New Methods for Solving Elliptic Equations, North-Holland, Amsterdam, 1967.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Reichel, L. (1987). Boundary collocation in Fejér points for computing eigenvalues and eigenfunctions of the Laplacian. In: Saff, E.B. (eds) Approximation Theory, Tampa. Lecture Notes in Mathematics, vol 1287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078903

Download citation

  • DOI: https://doi.org/10.1007/BFb0078903

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18500-0

  • Online ISBN: 978-3-540-47991-8

  • eBook Packages: Springer Book Archive