On a New Boundary Element Spectral Method

  • F. K. Hebeker
Part of the Progress in Scientific Computing book series (PSC, volume 7)


An efficient numerical algorithm for partial differential equations in complicated 3-D geometries is developed in case of viscous fluid flows. The algorithm consists essentially of a combination of a boundary element method (where the resulting linear algebraic system is solved efficiently with a multigrid procedure) and a spectral method to treat the nonhomogeneous part in the differential equations. Our investigations cover an exact mathematical foundation, a rigorous convergence analysis, and some 3-D numerical tests.


Boundary Element Boundary Element Method Boundary Integral Equation Stokes Problem Numerical Quadrature 
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. [1]
    Atkinson K.E., Solving Integral Equations on Surfaces in Space. G. Hämmerlin (ed.), Numerical Solution of Integral Equations, Basel 1985, 20–43.Google Scholar
  2. [2]
    Borchers W., Eine Fourier-Spektralmethode für das Stokes-Resolven-tenproblem. Submitted for publication, Univ. of Paderborn 1985, 13 pp.Google Scholar
  3. [3]
    Brebbia C.A. and Maier G. (eds.), Boundary Elements VII. 2 vols., Berlin 1985.Google Scholar
  4. [4]
    Duffy M.G., Quadrature over a Pyramid or Cube of Integrands with a Singularity at a Vertex. SIAM J. Numer. Anal. 19 (1982), 1260–1262.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Fischer T.M., An Integral Equation Procedure for the Exterior 3-D Slow Viscous Flow. Integral Equ. Oper. Th. 5 (1982), 490–505.MATHCrossRefGoogle Scholar
  6. [6]
    Fischer T.M. and Rosenberger R., A Boundary Integral Method for the Numerical Computation of the Forces Exerted on a Sphere in Viscous Incompressible Flows Near a Plane Wall. Preprint, Univ. of Darmstadt 1985, 30 pp.Google Scholar
  7. [7]
    Hackbusch W., Die schnelle Auflösung der Fredholm’schen Integralgleichung zweiter Art. Beitr. Nurner. Math. 9 (1982), 47–62.Google Scholar
  8. [8]
    Hebeker F.K., A Theorem of Faxén and the Boundary Integral Method for 3-D Viscous Incompressible Flows. Techn. Rep., Univ. of Paderborn 1982, 14 pp.Google Scholar
  9. [9]
    Hebeker F.K., A Boundary Integral Approach to Compute the 3-D Oseen Flow Past a Moving Body. M. Pandolfi and R. Piva (eds.), Numerical Methods in Fluid dynamics. Braunschweig 1984, 124–130.Google Scholar
  10. [10]
    Hebeker F.K., On a Multigrid Method for the Integral Equations of 3-D Stokes Flow. W. Hackbusch (ed.), Efficient Solution of Elliptic Systems. Braunschweig 1984, 67–73.Google Scholar
  11. [11]
    Hebeker F.K., Zur Randelemente-Methode in der 3-D Viskosen Strömungsmechanik. Habilitation Thesis, Univ. of Paderborn 1984, 127 pp.Google Scholar
  12. [12]
    Hebeker F.K., A Boundary Element Method for Stokes Equations in 3-D Exterior Domains. J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications. London 1985, 257–263.Google Scholar
  13. [13]
    Hebeker F.K., Efficient Boundary Element Methods for 3-D Exterior Viscous Flows. Submitted for publication, Univ. of Paderborn 1985, 35 pp.Google Scholar
  14. [14]
    Hebeker F.K., Efficient Boundary Element Methods for 3-D Viscous Flows. C.A. Brebbia and G. Maier (eds.), Boundary Elements VII. Berlin 1985, 9:37–44.Google Scholar
  15. [15]
    Hsiao G.C. and Kopp P. and Wendland W.L., A Collocation Method for some Integral Equations of the First Kind. Computing 25 (1980), 89–130.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Hsiao G.C. and Kopp P. and Wendland W.L., Some Applications of a Galerkin Collocation Method for Boundary Integral Equations of the First Kind. Math. Meth. Appl. Sci. 6 (1984), 280–325.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Hsiao G.C. and Kreß R., On an Integral Equation for the 2-D Exterior Stokes Problem. Appl. Numer. Math. 1 (1985), 77–93.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Hsiao G.C. and Wendland W.L. and Fischer T.M., Singular Perturbations for the Exterior 3-D Slow Viscous Flow Problem. J. Math. Anal. Appl., in press.Google Scholar
  19. [19]
    Ladyzhenskaja O.A., The Mathematical Theory of Viscous Incompressible Flow. New York 1969.Google Scholar
  20. [20]
    Nedelec J.C., Approximation des Equations Integrales en Mechanique et en Physique. Ecole Polytechnique Palaiseau 1977, 127 pp.Google Scholar
  21. [21]
    Novak Z., Use of the Multigrid Method for Laplacean Problems in 3-D. W. Hackbusch and U. Trottenberg (eds.), Multigrid Methods. Berlin 1982, 576–598.Google Scholar
  22. [22]
    Quarteroni A. and Maday Y., Spectral and Pseudospectral Approximations of the Navier Stokes Equations. SIAM J. Numer. Anal. 19 (1982), 761–780.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Schlichting H., Grenzschicht-Theorie. 8th ed., Karlsruhe 1982.MATHGoogle Scholar
  24. [24]
    Schwab Ch. and Wendland W.L., 3-D BEM and Numerical Integration. C.A. Brebbia and G. Maier (eds.), Boundary Elements VII. Berlin 1985, 13:85–101.Google Scholar
  25. [25]
    Scott L.R. and Johnson C., An Analysis of Quadrature Errors in second-kind Boundary Integral Equation Methods. Preprint, Univ. of Michigan (Ann Arbor), 47 pp.Google Scholar
  26. [26]
    Varnhorn W., Zur Numerik der Gleichungen von Navier-Stokes. Doctoral Thesis, Univ. of Paderborn, to appear.Google Scholar
  27. [27]
    Wendland W.L., Die Behandlung von Randwertaufgaben im ℝ3 mit Hilfe von Einfach- und Doppel Schichtpotential en. Numer. Math. 11 (1968), 380–404.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Wendland W.L., On the Asymptotic Convergence of some Boundary Element Methods. J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications. London 1982, 281–312.Google Scholar
  29. [29]
    Wendland W.L., Boundary Element Methods and Their Asymptotic Convergence. P. Filippi (ed.), Theoretical Acoustics and Numerical Techniques. CISM Courses and Lectures 277, Wien 1983, 135–216.Google Scholar
  30. [30]
    Wendland W.L., On some Mathematical Aspects of Boundary Element Methods for Elliptic Problems. J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications V. London 1985, 193–227.Google Scholar
  31. [31]
    Zhu J., A Boundary Integral Equation Method for the Stationary Stokes Problem in 3-D. C.A. Brebbia (ed.), Boundary Element Methods. Berlin 1983, 283–292.Google Scholar
  32. [32]
    Temam R., Navier Stokes Equations and Nonlinear Functional Analysis SIAM, Philadelphia 1983.Google Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • F. K. Hebeker

There are no affiliations available

Personalised recommendations