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

  • Lothar Reichel
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1287)


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BN]
    R.H.T. Bates and F.L. Ng, Point matching computation of transverse resonances, Int. J. Numer. Meth. Engng., 6 (1973), 155–168.CrossRefzbMATHGoogle Scholar
  2. [CH]
    R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Wiley, New York, 1953.Google Scholar
  3. [Cu]
    J.H. Curtiss, Transfinite diameter and harmonic polynomial interpolation, J. d’Analyse Math., 22 (1969), 371–389.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [Ei]
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [FHM]
    L. Fox, P. Henrici and C. Moler, Approximation and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal, 4 (1967), 89–102.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [Fi]
    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. [Ga]
    D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel, 1980.CrossRefzbMATHGoogle Scholar
  8. [Gu]
    M.H. Gutknecht, Numerical conformal mapping methods based on function conjugation, J. Comput. Appl. Math., 14 (1986), 31–77, in [T].CrossRefzbMATHMathSciNetGoogle Scholar
  9. [He]
    P. Henrici, Applied and Computational Complex Analysis, vol. 3, Wiley, New York, 1986.Google Scholar
  10. [HZ]
    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.CrossRefGoogle Scholar
  11. [KS]
    J.R. Kuttler and V.G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26 (1984), 163–193.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [Mo]
    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. [MP]
    C.B. Moler and L.E. Payne, Bounds for eigenvalues and eigenvectors of symmetric operators, SIAM J. Numer. Anal., 5 (1968), 64–70.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [Re1]
    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. [Re2]
    L. Reichel, On complex rational approximation by interpolation at preselected nodes, Complex Variables: Theory and Appl., 4 (1984), 63–87.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [Re3]
    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.CrossRefMathSciNetGoogle Scholar
  17. [Re4]
    L. Reichel, Numerical methods for analytic continuation and mesh generation, Constr. Approx., 2 (1986), 23–39.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [SH]
    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.CrossRefGoogle Scholar
  19. [T]
    L.N. Trefethen, ed., Numerical Conformal Mapping, J. Comput. Appl. Math., 14 (1986).Google Scholar
  20. [Ve]
    I.N. Vekua, New Methods for Solving Elliptic Equations, North-Holland, Amsterdam, 1967.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Lothar Reichel
    • 1
  1. 1.Department of MathematicsUniversity of KentuckyLexington

Personalised recommendations