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 Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2016, Volume 158, Book 1, Pages 40–50 (Mi uzku1350)

Polynomial interpolation of the function of two variables with large gradients in the boundary layers

Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Omsk, 644043 Russia

Abstract: The problem of interpolation of the function of two variables with large gradients in the boundary layers is investigated. It is assumed that the function has large gradients near the boundaries of a rectangular domain. Such function corresponds to the solution of the elliptic equation with a small parameter in the highest derivatives. It is known that the error of polynomial interpolation on a uniform grid for the function can be of the order of $O(1)$. It is suggested to use the two-dimensional Lagrange interpolation on the piecewise uniform Shishkin mesh, which is dense in the boundary layers. The Lagrange polynomial with $k_1$ interpolation nodes on $x$ and with $k_2$ interpolation nodes on $y$ is used. The error estimate which is uniform in the small parameter is obtained. Results of the numerical experiments are discussed.

Keywords: function of two variables, large gradients, polynomial interpolation, Shishkin mesh, error estimate.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-0658416-01-00727 This study was supported in part by the Russian Foundation for Basic Research (projects nos. 15-01-06584 and 16-01-00727).

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UDC: 519.65

Citation: A. I. Zadorin, N. A. Zadorin, “Polynomial interpolation of the function of two variables with large gradients in the boundary layers”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 158, no. 1, Kazan University, Kazan, 2016, 40–50

Citation in format AMSBIB
\Bibitem{ZadZad16} \by A.~I.~Zadorin, N.~A.~Zadorin \paper Polynomial interpolation of the function of two variables with large gradients in the boundary layers \serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki \yr 2016 \vol 158 \issue 1 \pages 40--50 \publ Kazan University \publaddr Kazan \mathnet{http://mi.mathnet.ru/uzku1350} \elib{https://elibrary.ru/item.asp?id=25848947} 

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This publication is cited in the following articles:
1. Tikhovskaya S.V., Zadorin A.I., “Analysis of Polynomial Interpolation of the Function of Two Variables With Large Gradients in the Parabolic Boundary Layers”, Application of Mathematics in Technical and Natural Sciences (Amitans'16), AIP Conference Proceedings, 1773, ed. Todorov M., Amer Inst Physics, 2016, 100008
2. S. V. Tikhovskaya, “Analysis of the numerical differentiation formulas of functions with large gradients”, Application of Mathematics in Technical and Natural Sciences, AIP Conf. Proc., 1895, ed. M. Todorov, Amer. Inst. Phys., 2017, UNSP 110010-1
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