Abstract
The problem of parabolic spline interpolation, according to Subbotin, of functions with large gradients in the boundary layer is considered. In the case of a uniform grid it is proved and in the case of the Shishkin mesh it is experimentally shown that with a parabolic spline interpolation of functions with large gradients in the exponential boundary layer, the error can unrestrictedly increase when the small parameter tends to zero and the number of grid nodes is fixed. An approximation process using parabolic splines with defect 1 is proposed; the error estimates are found to be uniform in the small parameter.
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Il’in, A.M., Difference Scheme for a Differential Equation with a Small Parameter at the Highest Derivative, Mat. Zam., 1969, vol. 6, no. 2, pp. 237–248.
Bahvalov, N.S., On Optimization of Solving Boundary Value Problems in the Presence of Boundary Layer, Zh. Vych. Mat. Mat. Fis., 1969, vol. 9, no. 4, pp. 841–890.
Shishkin, G.I., Setochnye approksimatsii singulyarno vozmushchennykh ellipticheskikh i parabolicheskikh uravnenii (Mesh Approximations of Singularly Perturbed Elliptic and Parabolic Equations), Yekaterinburg: UB RAS, 1992.
Shishkin, G.I., Approximation of Solutions to Singularly Perturbed Boundary Value Problems with Parabolic Boundary Layer, Zh. Vych. Mat. Mat. Fis., 1989, vol. 29, no. 7, pp. 963–977.
Ahlberg, J.H., Nilson, E.N., and Walsh, J.L., The Theory of Splines and Their Applications, New York: Academic Press, 1967.
De Boor, C., Prakticheskoe rukovodstvo po splainam (A Practical Guide to Splines), Moscow: Radio i Svyaz’, 1985.
Stechkin, S.B. and Subbotin, Yu.N., Splainy v vychislitel’noi matematike (Splines in Computational Mathematics), Moscow: Nauka, 1976.
Zav’yalov, Yu.S., Kvasov, B.I., and Miroshnichenko, V.L., Metody splain-funktsii (Methods of Spline Functions), Moscow: Nauka, 1980.
Zadorin, A.I., Method of Interpolation for a Boundary Layer Problem, Sib. Zh. Vych. Mat., 2007, vol. 10, no. 3, pp. 267–275.
Zadorin, A.I., Lagrange Interpolation and Newton–Cotes Formulas for Functions with a Boundary Layer Component on Piecewise Uniform Grids, Sib. Zh. Vych. Mat., 2015, vol. 18, no. 3, pp. 289–303.
Zadorin, A.I., Spline Interpolation of Functions with a Boundary Layer Component, Int. J. Numer. An.Mod., Series B, 2011, vol. 2, nos. 2/3, pp. 262–279.
Volkov, Yu.S., Interpolation with Even Degree Splines According to Subbotin and Marsden, Ukr. Mat. Zh., 2014, vol. 66, no. 7, pp. 891–908.
Miller, J.J.H., O’Riordan, E., and Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (rev. ed.), Singapore:World Scientific, 2012.
Linss, T., The Necessity of Shishkin Decompositions, Appl.Math. Lett., 2001, vol. 14, pp. 891–896.
Volkov, Yu.S., On Finding a Complete Interpolation Spline through B-Splines, Sib. El. Mat. Izv., 2008, vol. 5, pp. 334–338.
Volkov, Yu.S., Obtaining a Banded System of Equations in Complete Spline Interpolation Problem via BSplines Basis, Central Eur. J. Math., 2012, vol. 10, no. 1, pp. 352–356.
Blatov, I.A. and Kitaeva, E.V., Convergence of the BahvalovGridAdaptationMethod for Singularly Perturbed Boundary Value Problems, Sib. Zh. Vych. Mat., 2016, vol. 19, no. 1, pp. 43–55.
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Original Russian Text © I.A. Blatov, A.I. Zadorin, E.V. Kitaeva, 2017, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2017, Vol. 20, No. 2, pp. 131–144.
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Blatov, I.A., Zadorin, A.I. & Kitaeva, E.V. On the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer. Numer. Analys. Appl. 10, 108–119 (2017). https://doi.org/10.1134/S1995423917020021
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DOI: https://doi.org/10.1134/S1995423917020021