Abstract:
The interpolation problem of a one-variable function, which can be considered as a solution of a boundary value problem for an equation with a small parameter ε with a higher derivative is investigated. The application of the Lagrange interpolation for such a function on a uniform grid can result in serious errors. In the case of the Shishkin mesh, ε-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The ε-uniform error estimates of the Newton–Cotes formulas on such meshes are obtained. Numerical experiments have been carried out. The results obtained confirm the theoretical estimates.
Citation:
A. I. Zadorin, “The Lagrange interpolation and the Newton–Cotes formulas for functions with a boundary layer component on piecewise-uniform meshes”, Sib. Zh. Vychisl. Mat., 18:3 (2015), 289–303; Num. Anal. Appl., 8:3 (2015), 235–247
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\by A.~I.~Zadorin
\paper The Lagrange interpolation and the Newton--Cotes formulas for functions with a~boundary layer component on piecewise-uniform meshes
\jour Sib. Zh. Vychisl. Mat.
\yr 2015
\vol 18
\issue 3
\pages 289--303
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\crossref{https://doi.org/10.15372/SJNM20150304}
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\jour Num. Anal. Appl.
\yr 2015
\vol 8
\issue 3
\pages 235--247
\crossref{https://doi.org/10.1134/S1995423915030040}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84938585537}
Linking options:
https://www.mathnet.ru/eng/sjvm582
https://www.mathnet.ru/eng/sjvm/v18/i3/p289
This publication is cited in the following 17 articles:
A. I. Zadorin, “Two-dimensional interpolation of functions with large gradients in boundary layers”, Sib. elektron. matem. izv., 19:2 (2022), 688–697
A. I. Zadorin, N. A. Zadorin, “Lagrange interpolation and the Newton–Cotes formulas on a Bakhvalov mesh in the presence of a boundary layer”, Comput. Math. Math. Phys., 62:2 (2022), 347–358
A I Zadorin, “Approaches to constructing two-dimensional interpolation formulas in the presence of boundary layers”, J. Phys.: Conf. Ser., 2182:1 (2022), 012036
N. A. Zadorin, “Optimization of nodes of composite quadrature formulas in the presence of a boundary layer”, Sib. elektron. matem. izv., 18:2 (2021), 1201–1209
A. I. Zadorin, N. A. Zadorin, “Non-polynomial interpolation of functions with large gradients and its application”, Comput. Math. Math. Phys., 61:2 (2021), 167–176
N A Zadorin, S B Shagaev, “Two-grid method for the stationary Burgers equation”, J. Phys.: Conf. Ser., 1791:1 (2021), 012090
A I Zadorin, “New approaches to constructing quadrature formulas for functions with large gradients”, J. Phys.: Conf. Ser., 1901:1 (2021), 012055
A. I. Zadorin, “Optimization of nodes of newton-cotes formulas in the presence of an exponential boundary layer”, Iv International Scientific and Technical Conference Mechanical Science and Technology Update (Mstu-2020), Journal of Physics Conference Series, 1546, IOP Publishing Ltd, 2020, 012107
N A Zadorin, “Non-polynomial interpolation of functions in the presence of a boundary layer”, J. Phys.: Conf. Ser., 1441:1 (2020), 012179
I. A. Blatov, N. A. Zadorin, “Interpolyatsiya na setke Bakhvalova pri nalichii eksponentsialnogo pogranichnogo sloya”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 161, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2019, 497–508
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Approximation of a function and its derivatives on the basis of cubic spline interpolation in the presence of a boundary layer”, Comput. Math. Math. Phys., 59:3 (2019), 343–354
A. I. Zadorin, “The analysis of numerical differentiation formulas on the Shishkin mesh with of a boundary layer”, Num. Anal. Appl., 11:3 (2018), 193–203
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Cubic spline interpolation of functions with high gradients in boundary layers”, Comput. Math. Math. Phys., 57:1 (2017), 7–25
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer”, Num. Anal. Appl., 10:2 (2017), 108–119
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Parabolic spline interpolation for functions with large gradient in the boundary layer”, Siberian Math. J., 58:4 (2017), 578–590
A. I. Zadorin, N. A. Zadorin, “Polinomialnaya interpolyatsiya funktsii dvukh peremennykh s bolshimi gradientami v pogranichnykh sloyakh”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 158, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2016, 40–50
A. I. Zadorin, “Kvadraturnaya formula Gaussa na kusochno-ravnomernoi setke dlya funktsii s bolshimi gradientami v pogranichnom sloe”, Sib. elektron. matem. izv., 13 (2016), 101–110
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