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 
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  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.MathSciNetzbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Schlichting H., Grenzschicht-Theorie. 8th ed., Karlsruhe 1982.zbMATHGoogle 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.MathSciNetzbMATHCrossRefGoogle 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

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© Birkhäuser Boston 1987

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  • F. K. Hebeker

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