Cubic spline interpolation of functions with high gradients in boundary layers

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• 作者：I. A. Blatov ; A. I. Zadorin ; E. V. Kitaeva
• 关键词：singular perturbation ; boundary layer ; Shishkin mesh ; cubic spline ; modification ; error estimate
• 刊名：Computational Mathematics and Mathematical Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：57
• 期：1
• 页码：7-25
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Computational Mathematics and Numerical Analysis;
• 出版者：Pleiades Publishing
• ISSN：1555-6662
• 卷排序：57

文摘

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.