, 76:227 | Cite as

Optimal Quadratic and Cubic Spline Collocation on Nonuniform Partitions

  • C. C. ChristaraEmail author
  • Kit Sun  Ng


We develop optimal quadratic and cubic spline collocation methods for solving linear second-order two-point boundary value problems on non-uniform partitions. To develop optimal nonuniform partition methods, we use a mapping function from uniform to nonuniform partitions and develop expansions of the error at the nonuniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the spline collocation approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline collocation approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. Numerical results on a variety of problems, including a boundary-layer problem, and a nonlinear problem, verify the optimal convergence of the methods, even under more relaxed conditions than those assumed by theory.

AMS Subject Classiffications

65L10 65L20 65L50 65L60 65L70 65D05 65D07 


Spline collocation second-order two-point boundary value problem error bounds optimal order of convergence spline interpolation 


  1. Archer, D. 1977An O(h 4) cubic spline collocation method for quasilinear parabolic equationsSIAM J. Numer. Anal14620637CrossRefzbMATHMathSciNetGoogle Scholar
  2. Ascher, U., Christiansen, J., Russell, R. D. 1979A collocation solver for mixed order systems of boundary value problemsMath. Comp.33659679MathSciNetzbMATHGoogle Scholar
  3. Bader, G., Ascher, U. 1987A new basis implementation for a mixed order boundary value ODE solverSIAM J. Sci. Stat. Comp.8483500CrossRefMathSciNetzbMATHGoogle Scholar
  4. Celia, M. A., Gray, W. G.: Numerical methods for differential equations. Prentice Hall 1992.Google Scholar
  5. Christara, C. C. 1994Quadratic spline collocation methods for elliptic partial differential equationsBIT343361CrossRefzbMATHMathSciNetGoogle Scholar
  6. Christara, C., Ng, K. S.: Adaptive techniques for spline collocation. Computing 76, 259–277 (2006) (this issue).Google Scholar
  7. Daniel, J. W., Swartz, B. K. 1975Extrapolated collocation for two-point boundary-value problems using cubic splinesJ. Inst. Maths Appl.16161174MathSciNetCrossRefzbMATHGoogle Scholar
  8. Boor, C., Swartz, B. 1973Collocation at Gaussian pointsSIAM J. Numer. Anal.10582606MathSciNetzbMATHGoogle Scholar
  9. Fyfe, D. J. 1969The use of cubic splines in the solution of two-point boundary value problemsComput. J.12188192CrossRefzbMATHMathSciNetGoogle Scholar
  10. Houstis, E. N., Christara, C. C., Rice, J. R. 1988Quadratic-spline collocation methods for two-point boundary value problemsInt. J. Numer. Meth. Eng.26935952CrossRefMathSciNetzbMATHGoogle Scholar
  11. Houstis, E. N., Mitchell, W. F., Rice, J. R. 1985Collocation software for second-order elliptic partial differential equationsACM Trans. Math. Soft.11379412MathSciNetzbMATHGoogle Scholar
  12. Houstis, E. N., Vavalis, E. A., Rice, J. R. 1988Convergence of an O(h 4) cubic spline collocation method for elliptic partial differential equationsSIAM J. Numer. Anal.255474CrossRefMathSciNetzbMATHGoogle Scholar
  13. Irodotou-Ellina, M., Houstis, E. N. 1988AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problemsBIT28288301CrossRefMathSciNetzbMATHGoogle Scholar
  14. Ng, K. S.: Spline Collocation on adaptive grids and non-rectangular regions. PhD thesis, Department of Computer Science, University of Toronto, Toronto, Ontario, Canada, 2005 Scholar
  15. Prenter, P. M.: Splines and variational methods. New York: Wiley 1975.Google Scholar
  16. Russell, R. D., Shampine, L. F. 1971A collocation method for boundary value problemsNumer. Math.19128MathSciNetGoogle Scholar
  17. Zhu, Y.: Optimal quartic spline collocation methods for fourth order two-point boundary value problems. M.Sc. thesis, Department of Computer Science, University of Toronto, Toronto, Ontario, Canada., April 2001.Google Scholar

Copyright information

© Springer-Verlag Wien 2005

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations