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Error analysis for spline collocation methods with application to knot selection

Authors: J. Christiansen and R. D. Russell
Journal: Math. Comp. 32 (1978), 415-419
MSC: Primary 65L10
MathSciNet review: 0494963
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Abstract: Some collocation schemes used to solve mth order ordinary differential equations are known to display superconvergence at the mesh points. Here we show that some such schemes have additional superconvergence points for the approximate solution and all of its derivatives. Using such points, we argue that a mesh selection scheme introduced by Dodson can be expected to perform well under general circumstances. A numerical example is given to demonstrate the new superconvergence results.

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  • [1] Carl de Boor, Good approximation by splines with variable knots. II, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 12–20. Lecture Notes in Math., Vol. 363. MR 0431606
  • [2] Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328,
  • [3] D. J. DODSON, Optimal Order Approximation by Spline Functions, Ph.D. Thesis, Purdue Univ., 1972.
  • [4] R. D. Russell and J. Christiansen, Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal. 15 (1978), no. 1, 59–80. MR 471336,
  • [5] A. B. WHITE, JR., On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems, Report 112, Univ. of Texas, Center Numer. Anal., 1976.

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Article copyright: © Copyright 1978 American Mathematical Society