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Numerische Mathematik

, Volume 66, Issue 1, pp 67–95 | Cite as

On quadrature methods for the double layer potential equation over the boundary of a polyhedron

  • A. Rathsfeld
Article

Summary

In this paper we consider a quadrature method for the solution of the double layer potential equation corresponding to Laplace's equation in a threedimensional polyhedron. We prove the stability for our method in case of special triangulations over the boundary of the polyhedron. The assumptions imposed on the triangulations are analogous to those appearing in the one-dimensional case. Finally, we establish the rates of convergence and discuss the effect of mesh refinement near the corners and edges of the polyhedron.

Mathematics Subject Classification (1991)

45L10 65R20 

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References

  1. 1.
    Adolfsson, V., Goldberg, M., Jawerth, B. Lennerstad, H. (1992): Localized Galerkin estimates for boundary integral equations on Lipschitz domains SIAM J. Math. Anal.23 (5), 1356–1374Google Scholar
  2. 2.
    Atkinson, K.E., Chien, D. (1992): Piecewise polynomial collocation for boundary integral equations. Report No. 29. Department of Mathematics, University of Iowa, Iowa CityGoogle Scholar
  3. 3.
    Angell, T.S., Kleinman, R.E., Kral, J. (1988): Layer potentials on boundaries with corners and edges. Cas. Pestovani Mat.113(4), 387–402Google Scholar
  4. 4.
    Baker, C.T.H. (1977): Numerical treatment of integral equations. Claredon Press, OxfordGoogle Scholar
  5. 5.
    Chandler, G.A., Graham, I.G. (1988): High order methods for linear functionals of solutions of second kind integral equations. SIAM J. Numer. Anal.25, 1118–1137Google Scholar
  6. 6.
    Ciarlet, P.G. (1979): The finite element method for elliptic problems. North-Holland. AmsterdamGoogle Scholar
  7. 7.
    Costabel, M., Penzel, F., Schneider, R. (1992): Error analysis of a boundary element collocation method for a screen problem in ℝ3. Math. Comput58, 575–586Google Scholar
  8. 8.
    Dahlberg, B.E.J., Verchota, G. (1990): Galerkin methods for the boundary integral equations of elliptic equations in non-smooth domains. Contemp. Math.107, 39–60Google Scholar
  9. 9.
    Elschner, J. (1990): On spline approximation for a class of non-compact integral equations. Math. Nachr.146, 271–321Google Scholar
  10. 10.
    Elschner, J. (1992): The double layer potential operator over polyhedral domains II: Spline Galerkin methods Math. Meth. Appl. Sci.15, 23–37Google Scholar
  11. 11.
    Engels, H. (1988): Numerical quadrature and cubature. Academic Press, LondonGoogle Scholar
  12. 12.
    Gohberg, I., Krupnik, N. (1979): Einführung in die Theorie der eindimensionalen singulaeren Integraloperatoren. Birkhäuser, Basel, StuttgartGoogle Scholar
  13. 13.
    Kleemann, B., Rathsfeld, A. (1993): Nyström's method and iterative solvers for the solution of the double layer potential equation over polyhedral domains, IAAS-Berlin Preprint 36 BerlinGoogle Scholar
  14. 14.
    Kleinman, R., Wendland, W.L. (1977): On Neumann's method for the exterior Neumann problem for the Helmholtz equation. J. Math. Anal. Appl.57, 170–202Google Scholar
  15. 15.
    Kozak, A.V. (1974): A local principle in the theory of projection methods. Dokl. Akad. Nauk 212(6), 1287–1289 (russian), Soviet Dokl.14, 1580–1583Google Scholar
  16. 16.
    Kral, J. (1980): Integral operators in potential theory. LNM 823, Springer, Berlin Heidelberg New YorkGoogle Scholar
  17. 17.
    Kral, J., Wendland, W. (1988): On the applicability of the Fredholm-Radon method in potential theory and the panel method. In: J. Ballmann, R. Eppler, W. Hackbusch eds. Notes on Numerical Fluid Mechanics,21, 120–136, ViehwegGoogle Scholar
  18. 18.
    Kral, J., Wendland, W., (1986): Some examples concerning applicability of the Fredholm-Radon method in potential theory. Apl. Mat.31, 293–308Google Scholar
  19. 19.
    Mazya, V.G. (1991): Boundary integral equations. In: V.G. Mazya, S.M. Nikol'skiî eds., Encyclopaedia of Math. Sciences Analysis IV.27, pp. 127–233, Springer, Berlin Heidelberg New YorkGoogle Scholar
  20. 20.
    Penzel, F. (1990): Error estimates for a discretized Galerkin method for a boundary integral equation in two dimension. TH Darmstadt, Preprint 1276, DarmstadtGoogle Scholar
  21. 21.
    von Petersdorff, T. (1989): Randwertprobleme der Elastizitaetstheorie für Polyeder-Singularitäten und Approximation mit Randelementmethoden. Thesis, TH DarmstadtGoogle Scholar
  22. 22.
    Prößdorf, S. (1984): Ein Lokalisierungsprinzip in der Theorie der Splineapproximationen und einige Anwendungen. Math. Nachr.119, 239–255Google Scholar
  23. 23.
    Prößdorf, S., Rathsfeld, A. (1984): A spline collocation method for singular integral equations with piecewise continuous coefficients. Int. Equ. Op. Theory7, 536–560Google Scholar
  24. 24.
    Prößdorf, S., Schneider, R. (1991): A spline collocation method for multi-dimensional strongly elliptic pseudodifferential operators of order zero. Int. Equ. Op. Theory14, 399–435Google Scholar
  25. 25.
    Prößdorf, S., Schneider, R. (1992): Spline approximation methods for multi-dimensional periodic pseudodifferential equations. Int. Equ. Op. Theory15, 626–672Google Scholar
  26. 26.
    Rathsfeld, A. (1991): On the stability of quadrature methods for the double layer potential equation over the boundary of a polyhedron. Karl-Weierstrass-Institut für Mathematik, Preprint P-MATH-09/91, BerlinGoogle Scholar
  27. 27.
    Rathsfeld, A. (1992): The invertibility of the double layer potential operator in the space of continuous functions defined on a polyhedron. The panel method. Applicable Analysis45, 135–177Google Scholar
  28. 28.
    Silbermann, B. (1980): Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren. Math. Nachr.104, 137–145Google Scholar
  29. 29.
    Wendland, W. (1968): Die Behandlung von Randwertaufgaben im ℝ3 mit Hilfe von Einfach-und Doppelschichtpotentialen. Numer. Math.11, 380–404Google Scholar
  30. 30.
    Wendland, W. (1983): Boundary element methods and their asymptotic convergence. In: P. Filippi, ed., Theoretical acoustics and numerical techniques. CISM Courses vol. 277, pp. 135–216, Springer, Berlin Heidelberg New YorkGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • A. Rathsfeld
    • 1
  1. 1.Institut für Angewandte Analysis und StochastikBerlinGermany

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