Geodesic
Interpolation formulas for functions with large gradients in the boundary layer and their application
Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 3, pp. 377-384.

Voir la notice de l'article provenant de la source Math-Net.Ru

Interpolation of functions on the basis of Lagrange's polynomials is widely used. However in the case when the function has areas of large gradients, application of polynomials of Lagrange leads to essential errors. It is supposed that the function of one variable has the representation as a sum of regular and boundary layer components. It is supposed that derivatives of a regular component are bounded to a certain order, and the boundary layer component is a function, known within a multiplier; its derivatives are not uniformly bounded. A solution of a singularly perturbed boundary value problem has such a representation. Interpolation formulas, which are exact on a boundary layer component, are constructed. Interpolation error estimates, uniform in a boundary layer component and its derivatives are obtained. Application of the constructed interpolation formulas to creation of formulas of the numerical differentiation and integration of such functions is investigated.
Keywords: function of one variable, boundary layer component, formulas of numerical differentiation, error estimate.
Mots-clés : nonpolynomial interpolation, quadrature formulas 
@article{MAIS_2016_23_3_a15,
     author = {A. I. Zadorin},
     title = {Interpolation formulas for functions with large gradients in the boundary layer and their application},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {377--384},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2016_23_3_a15/}
}
TY  - JOUR
AU  - A. I. Zadorin
TI  - Interpolation formulas for functions with large gradients in the boundary layer and their application
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2016
SP  - 377
EP  - 384
VL  - 23
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2016_23_3_a15/
LA  - en
ID  - MAIS_2016_23_3_a15
ER  - 
%0 Journal Article
%A A. I. Zadorin
%T Interpolation formulas for functions with large gradients in the boundary layer and their application
%J Modelirovanie i analiz informacionnyh sistem
%D 2016
%P 377-384
%V 23
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2016_23_3_a15/
%G en
%F MAIS_2016_23_3_a15
A. I. Zadorin. Interpolation formulas for functions with large gradients in the boundary layer and their application. Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 3, pp. 377-384. http://geodesic.mathdoc.fr/item/MAIS_2016_23_3_a15/

[1] Zadorin A. I., “Method of interpolation for a boundary layer problem”, Suberian journal of numerical mathematics, 10:3 (2007), 267–275 (in Russian)

[2] Shishkin G. I., Discrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Branch, Ekaterinburg, 1992 (in Russian)

[3] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, Revised Edition, World Scientific, Singapore, 2012 | MR | Zbl

[4] Zadorin A. I., Zadorin N. A., “Interpolation formula for functions with a boundary layer component and its application to derivatives calculation”, Siberian Electronic Mathematical Reports, 9 (2012), 445–455 | MR | Zbl

[5] Bakhvalov N. S., Zhidkov N. P., Kobel'kov G. M., Numerical Methods, Nauka, M., 1987 (in Russian) | MR | Zbl

[6] Zadorin A. I., Zadorin N. A., “Spline interpolation on a uniform grid for functions with a boundary-layer component”, Computational Mathematics and Mathematical Physics, 50:2 (2010), 211–223 | DOI | MR | Zbl

[7] Zadorin A. I., Zadorin N. A., “Quadrature formulas for functions with a boundary-layer component”, Computational Mathematics and Mathematical Physics, 51:11 (2011), 1837–1846 | DOI | MR | Zbl

[8] Zadorin A. I., Zadorin N. A., “An analogue of the four-point Newton-Cotes formula for a function with a boundary-Layer Component”, Numerical Analysis and Applications, 6:4 (2013), 268–278 | DOI | MR | Zbl

[9] Zadorin A., Zadorin N., “Quadrature formula with five nodes for functions with a boundary layer component”, Lect. Notes in Comput. Sci., 8236, Springer, Berlin, 2013, 540–546 | DOI | MR | Zbl

[10] Zadorin A. I., Zadorin N. A., “Analogue of Newton-Cotes formulas for numerical integration of functions with a boundary-layer component”, Computational Mathematics and Mathematical Physics, 56:3 (2016), 358–366 | DOI | Zbl