Geodesic
The Lagrange interpolation and the Newton–Cotes formulas for functions with a boundary layer component on piecewise-uniform meshes
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 3, pp. 289-303 
A. I. Zadorin. The Lagrange interpolation and the Newton–Cotes formulas for functions with a boundary layer component on piecewise-uniform meshes. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 3, pp. 289-303. http://geodesic.mathdoc.fr/item/SJVM_2015_18_3_a3/
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     title = {The {Lagrange} interpolation and the {Newton{\textendash}Cotes} formulas for functions with a~boundary layer component on piecewise-uniform meshes},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\varepsilon$ 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, $\varepsilon$-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The $\varepsilon$-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.

[1] Bakhvalov N. S., Zhidkov N. P., Kobel'kov G. M., Chislennye metody, Nauka, Moskva, 1987 | MR

[2] Shishkin G. I., Setochnye approksimatsii singulyarno vozmushchennykh ellipticheskikh i parabolicheskikh uravnenij, UrO RAN, Ekaterinburg, 1992

[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., “Metod interpolyatsii dlya zadachi s pogranichnym sloem”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 10:3 (2007), 267–275

[5] Zadorin A. I., Zadorin N. A., “Splajn-interpolyatsiya na ravnomernoj setke funktsii s pogranslojnoj sostavlyayushchej”, Zhurn. vychisl. matem. i mat. fiziki, 50:2 (2010), 221–233 | MR | Zbl

[6] 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

[7] Lins T., “The necessity of Shishkin decompositions”, Applied Mathematics Letters, 14 (2001), 891–896 | DOI | MR | Zbl

[8] Kornev A. A., Chizhonkov E. V., Uprazhneniya po chislennym metodam, Chast' 2, MGU, M., 2003

[9] Il'in V. P., Chislennyj analiz, Izd-vo IVMiMG SO RAN, Novosibirsk, 2004

[10] Zadorin A. I., Zadorin N. A., “Kvadraturnye formuly dlya funktsij s pogranslojnoj sostavlyayushchej”, Zhurn. vychisl. matem. i mat. fiziki, 51:11 (2011), 1952–1962 | MR | Zbl

[11] Zadorin A. I., Zadorin N. A., “Analog formuly N'yutona–Kotesa s chetyr'mya uzlami dlya funktsii s pogranslojnoj sostavlyayushchej”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 16:4 (2013), 313–323 | Zbl

[12] Zadorin A., Zadorin N., “Quadrature Formula with Five Nodes for Functions with a Boundary Layer Component”, Numerical Analysis and its Applications, Lect. Notes in Comput. Sci., 8236, Springer, Berlin, 2013, 540–546 | DOI | MR | Zbl

[13] Zadorin A. I., Zadorin N. A., “Formula Simpsona i ee modifikatsii dlya funktsii s pogranslojnoj sostavlyayushchej”, Sibirskie elektronnye matematicheskie izvestiya, 11 (2014), 258–267

[14] Vulanovic R., “A priori meshes for singularly perturbed quasilinear two-point boundary value problems”, IMA J. Numer. Anal., 21:1 (2001), 349–366 | DOI | MR | Zbl

[15] Shishkin G. I., “Uluchshennaya kusochno-ravnomernaya setka dlya singulyarno vozmushchennogo ellipticheskogo uravneniya reaktsii-diffuzii”, Tr. Instituta matematiki i mekhaniki UrO RAN, 9, no. 2, 2003, 172–179 | MR | Zbl