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, 76:227 | Cite as

Optimal Quadratic and Cubic Spline Collocation on Nonuniform Partitions

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

Abstract

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 

Keywords

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