Numerische Mathematik

, Volume 62, Issue 1, pp 511–538 | Cite as

Spline collocation for strongly elliptic equations on the torus

  • Martin Costabel
  • William McLean


We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].

Mathematics Subject Classification (1991)

65R20 (35S15 41A15 41A63 65N35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agranovich, M.S. (1985): On elliptic pseudodifferential operators on a closed curve. Trans. Moscow Math. Soc.47, 23–74Google Scholar
  2. 2.
    Arnold, D.N., Saranen, J. (1984): On the asymptotic convergence of spline collocation methods for partial differential equations. SIAM J. Numer. Anal.21, 459–472Google Scholar
  3. 3.
    Arnold, D.N., Wendland, W.L. (1983): On the asymptotic convergence of collocation methods. Math. Comp.41, 349–381Google Scholar
  4. 4.
    Arnold, D.N., Wendland, W.L. (1985): The convergence of spline collocation for strongly elliptic equations on curves. Numer. Math.47, 317–341Google Scholar
  5. 5.
    Atkinson, K.E. (1990): A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions. In: M. Goldberg, ed., Numerical Solution of Integral Equations. Plenum Press, New YorkGoogle Scholar
  6. 6.
    Baker, C.T.H. (1977): The Numerical Treatment of Integral Equations. Clarendon Press, OxfordGoogle Scholar
  7. 7.
    Brebbia, C.A., Telles, J.C.F., Wrobel, L.C. (1984): Boundary Element Techniques. Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. 8.
    Chazarain, J., Piriou, A. (1982): Introduction to the Theory of Linear Partial Differential Equations. North-Holland, AmsterdamGoogle Scholar
  9. 9.
    Costabel, M., McLean, W. (1992): Global symbols of boundary integral operators on a torus. Proc. Centre Math. Anal. Austral. Nat. Univ. (1990) (to appear)Google Scholar
  10. 10.
    Costabel, M., Penzel, F., Schneider, R. (1992): Error analysis of a boundary element collocation method for a screen problem in ℝ3. THD-Preprint 1284, Technische Hochschule Darmstadt January 1989. Math. Comput. (to appear)Google Scholar
  11. 11.
    Costabel, M., Wendland, W.L. (1986): Strong ellipticity of boundary integral operators. J. Reine Angew. Math.372, 39–63Google Scholar
  12. 12.
    de Hoog, F.R. (1973): Product Integration Techniques for the Numerical Solution of Integral Equations. Ph.D. thesis, Australian National University, CanberraGoogle Scholar
  13. 13.
    Dieudonné, J. (1978): Eléments d'Analyse, vol. 8. Gauthier-Villars, ParisGoogle Scholar
  14. 14.
    Hsiao, G.C., Prößdorf, S. (1992): On spline collocation for multidimensional singular integral equations. To appearGoogle Scholar
  15. 15.
    McLean, W. (1989): Periodic pseudodifferential operators and periodic function spaces. Applied Mathematics Preprint AM89/10, University of New South Wales, SydneyGoogle Scholar
  16. 16.
    McLean, W. (1991): Local and global descriptions of periodic pseudodifferential operators. Math. Nachr.150, 151–161Google Scholar
  17. 17.
    Nitsche, J., Schatz, A. (1974): Interior estimates for Ritz-Galerkin methods. Math. Comput.28, 937–958Google Scholar
  18. 18.
    Petersen, B.E. (1983): Introduction to the Fourier Transform and Pseudodifferential Operators. Pitman, BostonGoogle Scholar
  19. 19.
    Prößdorf, S. (1989): Numerische Behandlung singulärer Integralgleichungen. Z. angew. Math. Mech.69, T5-T13Google Scholar
  20. 20.
    Prößdorf, S., Rathsfeld, A. (1984): A spline collocation method for singular integral equations with piecewise continuous coefficients. Integral Equations Oper. Theory7, 536–560Google Scholar
  21. 21.
    Prößdorf, S., Schneider, R. (1981): A finite element collocation method for singular integral equations. Math. Nachr.100, 33–60Google Scholar
  22. 22.
    Prößdorf, S., Schneider, R. (1990): Spline approximation methods for multidimensional periodic pseudodifferential equations. THD-Preprint 1341, Technische Hochschule DarmstadtGoogle Scholar
  23. 23.
    Prößdorf, S., Schneider, R. (1991): A spline collocation method for multidimensional strongly elliptic pseudodifferential operators of order zero. Integral Equations Oper. Theory14, 399–435Google Scholar
  24. 24.
    Saranen, J. (1988): The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane. Numer. Math.53, 499–512Google Scholar
  25. 25.
    Saranen, J., Wendland, W.L. (1985): On the asymptotic convergence of collocation methods with spline functions of even degree. Math. Comput.171, 91–108Google Scholar
  26. 26.
    Schneider, R. (1989): Stability of a collocation method for strongly elliptic multidimensional singular integral equations. Z. Anal. Anwendungen8, 361–376Google Scholar
  27. 27.
    Sloan, I.H. (1988): A quadrature-based approach to improving the collocation method. Numer. Math.54, 41–56Google Scholar
  28. 28.
    Sloan, I.H., Wendland, W.L. (1992): A quadrature-based approach to improving the collocation method for splines of even degree. Z. Anal. Anwend. (to appear)Google Scholar
  29. 29.
    Wendland, W.L. (1987): Strongly elliptic boundary integral equations. In: A. Iserles, M.J.D. Powell, eds., The State of the Art in Numerical Analysis. Clarendon Press, Oxford, pp. 511–562Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Martin Costabel
    • 1
  • William McLean
    • 2
  1. 1.Département de MathématiquesUniversité Bordeaux 1Talence CédexFrance
  2. 2.School of MathematicsUniversity of New South WalesSydneyAustralia

Personalised recommendations