Geodesic
About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 2, pp. 131-144.

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

A problem of the Subbotin parabolic spline-interpolation of functions with large gradients in the boundary layer is considered. In the case of a uniform grid it has been proved and in the case of the Shishkin grid it has been experimentally shown that with a parabolic spline-interpolation of functions with large gradients the error in the exponential boundary layer can unrestrictedly increase with a fixed number of grid nodes. A modified parabolic spline has been constructed. Estimates of the interpolation error of the constructed spline don't depend from a small parameter. 
@article{SJVM_2017_20_2_a1,
     author = {I. A. Blatov and A. I. Zadorin and E. V. Kitaeva},
     title = {About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {131--144},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a1/}
}
TY  - JOUR
AU  - I. A. Blatov
AU  - A. I. Zadorin
AU  - E. V. Kitaeva
TI  - About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2017
SP  - 131
EP  - 144
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a1/
LA  - ru
ID  - SJVM_2017_20_2_a1
ER  - 
%0 Journal Article
%A I. A. Blatov
%A A. I. Zadorin
%A E. V. Kitaeva
%T About the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2017
%P 131-144
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a1/
%G ru
%F SJVM_2017_20_2_a1
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. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 2, pp. 131-144. http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a1/

[1] Ilin A. M., “Raznostnaya skhema dlya differentsialnogo uravneniya s malym parametrom pri starshei proizvodnoi”, Matem. zametki, 6:2 (1969), 237–248 | MR | Zbl

[2] Bakhvalov N. S., “K optimizatsii metodov resheniya kraevykh zadach pri nalichii pogranichnogo sloya”, Zhurn. vychisl. matem. i mat. fiziki, 9:4 (1969), 841–859 | MR | Zbl

[3] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[4] Shishkin G. I., “Approksimatsiya reshenii singulyarno vozmuschennykh kraevykh zadach s parabolicheskim pogranichnym sloem”, Zhurn. vychisl. matem. i mat. fiziki, 29:7 (1989), 963–977 | MR | Zbl

[5] Ahlberg J. H., Nilson E. N., Walsh J. L., The Theory of Splines and their Applications, Academic Press, New York, 1967 | MR | Zbl

[6] De Bor K., Prakticheskoe rukovodstvo po splainam, Radio i svyaz, M., 1985 | MR

[7] Stechkin S. B., Subbotin Yu. N., Splainy v vychislitelnoi matematike, Nauka, M., 1976 | MR

[8] Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L., Metody splain-funktsii, Nauka, M., 1980 | MR

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

[10] Zadorin A. I., “Interpolyatsiya Lagranzha i formuly Nyutona–Kotesa dlya funktsii s pogransloinoi sostavlyayuschei na kusochno-ravnomernykh setkakh”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 18:3 (2015), 289–303 | DOI | MR | Zbl

[11] Zadorin A. I., “Spline interpolation of functions with a boundary layer component”, Inter. J. Numer. Anal. Mod. Series B, 2:2–3 (2011), 262–279 | MR | Zbl

[12] Volkov Yu. S., “Interpolyatsiya splainami chetnoi stepeni po Subbotinu i po Marsdenu”, Ukrainskii matematicheskii zhurnal, 66:7 (2014), 891–908 | Zbl

[13] 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

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

[15] Volkov Yu. S., “O nakhozhdenii polnogo interpolyatsionnogo splaina cherez B-splainy”, Sibirskie elektronnye matematicheskie izvestiya, 5 (2008), 334–338 | MR | Zbl

[16] Volkov Yu. S., “Obtaining a banded system of equations in complete spline interpolation problem via B-splines basis”, Central Europen J. of Mathematics, 10:1 (2012), 352–356 | DOI | MR | Zbl

[17] Blatov I. A., Kitaeva E. V., “Skhodimost metoda adaptatsii setok N. S. Bakhvalova dlya singulyarno vozmuschennykh kraevykh zadach”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 19:1 (2016), 47–59 | DOI | MR | Zbl