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Computing

, Volume 76, Issue 3–4, pp 259–277 | Cite as

Adaptive Techniques for Spline Collocation

  • C. C. ChristaraEmail author
  • Kit  Sun  Ng
Article

Abstract

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.

AMS Subject Classiffications

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

Keywords

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

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Copyright information

© Springer-Verlag Wien 2005

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

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