This article is cited in 21 papers

Cubic spline interpolation of functions with high gradients in boundary layers

I. A. Blatov^a, A. I. Zadorin^b, E. V. Kitaeva<sup>c

a</sup> Volga State University of Telecommunications and Informatics, Samara, Russia
^b Sobolev Institute of Mathematics (Omsk Branch), Siberian Branch, Russian Academy of Sciences, Omsk, Russia
^c Samara State University, Samara, Russia

Abstract: The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes $N$ is fixed. A modified cubic interpolation spline is proposed, for which $O((\ln N/N)^4)$ error estimates that are uniform with respect to the small parameter are obtained.

Key words: singular perturbation, boundary layer, Shishkin mesh, cubic spline, modification, error estimate.

UDC: 519.652.3

Received: 03.02.2016
Revised: 31.03.2016

DOI: 10.7868/S0044466917010057

 English version: Computational Mathematics and Mathematical Physics, 2017, 57:1, 7–25

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