Cubic spline interpolation of functions with high gradients in boundary layers

I. A. Blatov   ; A. I. Zadorin   ; E. V. Kitaeva

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 1, pp. 9-28

Résumé

The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes $N$ is fixed. A modified cubic interpolation spline is proposed, for which $O((\ln N/N)^4)$ error estimates that are uniform with respect to the small parameter are obtained.

Bibliographie

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

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

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

[4] Ahlberg J. H., Nilson E. N., Walsh J. L., The theory of splines and their applications, Academic Press, New York, 1967 | MR | Zbl

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

[6] Zadorin A. I., “Metod interpolyatsii dlya zadachi s pogranichnym sloem”, Sibirsk. zh. vychisl. matem., 10:3 (2007), 267–275

[7] Zadorin A. I., Guryanova M. V., “Analogue of a cubic spline for a function with a boundary layer component”, Proc. Fifth Conference on Finite Difference Methods: Theory and Applications (2010), Rousse University, Rousse, 2011, 166–173

[8] Zadorin A. I., “Interpolyatsiya Lagranzha i formuly Nyutona–Kotesa dlya funktsii s pogransloinoi sostavlyayuschei na kusochno-ravnomernykh setkakh”, Sibirsk. zh. vychisl. matem., 18:3 (2015), 289–303

[9] Zmatrakov N. L., “Skhodimost interpolyatsionnogo protsessa dlya parabolicheskikh i kubicheskikh splainov”, Tr. MIAN, 138, 1975, 71–93 | Zbl

[10] Zmatrakov N. L., “Neobkhodimoe uslovie skhodimosti interpolyatsionnykh parabolicheskikh i kubicheskikh splainov”, Matem. zametki, 19:2 (1976), 165–178 | Zbl

[11] 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

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

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

[14] Demko S., “Inverses of band matrices and local convergence of spline projections”, SIAM J. Numer. Anal., 14:4 (1977), 616–619 | DOI | MR | Zbl

[15] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984

[16] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978

[17] Blatov I. A., “O metodakh nepolnoi faktorizatsii dlya sistem s razrezhennymi matritsami”, Zh. vychisl. matem. i matem. fiz., 33:6 (1993), 819–836 | Zbl

[18] Volkov Yu. S., “O nakhozhdenii polnogo interpolyatsionnogo splaina cherez V-splainy”, Sibirsk. elektronnye matem. izvestiya, 5 (2008), 334–338 | Zbl

[19] Blatov I. A., Kitaeva E. V., “Skhodimost metoda adaptatsii setok N. S. Bakhvalova dlya singulyarno vozmuschennykh kraevykh zadach”, Sibirsk. zh. vychisl. matem., 19:1 (2016), 43–55 | Zbl