Geodesic
Cubic spline interpolation of functions with high gradients in boundary layers
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 1, pp. 9-28 
I. A. Blatov; A. I. Zadorin; E. V. Kitaeva. Cubic spline interpolation of functions with high gradients in boundary layers. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 1, pp. 9-28. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_1_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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