Geodesic
Interpolation of the function of two variables with large gradients in the boundary layers
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 2, pp. 55-67 Cet article a éte moissonné depuis la source Math-Net.Ru

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The problem of interpolation for the function of two variables with large gradients in the boundary layers is investigated. It is suggested that the function can be represented as a sum of a regular component with bounded derivatives up to some order and of two boundary layer components. The boundary layer components are known, but their coefficients are uncertain. Such representation is typical for the solution of a singular perturbed elliptic problem. A two-dimensional interpolation formula, which is exact on the boundary layer components, is deduced. The formula has the arbitrary number of nodes in each direction. The accuracy estimate, which is uniform in gradients of the interpolated function in the boundary layers, is proved. The results of numerical experiments are provided.
Keywords: function of two variables, boundary layer component
Mots-clés : large gradients, non-polynomial interpolation, error estimation. 
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     title = {Interpolation of the function of two variables with large gradients in the boundary layers},
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A. I. Zadorin. Interpolation of the function of two variables with large gradients in the boundary layers. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 2, pp. 55-67. http://geodesic.mathdoc.fr/item/UZKU_2015_157_2_a4/

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

[2] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp. | MR

[3] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992, 233 pp.

[4] 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, 176 pp. | MR | Zbl

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

[6] Zadorin A. I., Zadorin N. A., “Splain-interpolyatsiya na ravnomernoi setke funktsii s pogransloinoi sostavlyayuschei”, Zhurn. vychisl. matem. i matem. fiziki, 50:2 (2010), 221–233 | MR | Zbl

[7] Zadorin A. I., “Spline interpolation of functions with a boundary layer component”, Int. J. Numer. Anal. Model. Ser. B, 2:2–3 (2011), 262–279 | DOI | MR | Zbl

[8] Zadorin A. I., Zadorin N. A., “Interpolation formula for functions with a boundary layer component and its application to derivatives calculation”, Sib. Electron. Math. Rep., 9 (2012), 445–455 | MR

[9] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987, 600 pp. | MR

[10] Blatov I. A., Zadorin N. A., “Analiz interpolyatsionnoi formuly, tochnoi na pogransloinoi sostavlyayuschei interpoliruemoi funktsii”, Nauka i mir, 2015, no. 2(18), 13–17

[11] Kornev A. A., Chizhonkov E. V., Uprazhneniya po chislennym metodam, Chast dva, Mosk. gos. un-t, M., 2003, 200 pp.

[12] Roos H. G., Stynes M., Tobiska L., Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow problems, Springer, Berlin, 2008, 348 pp. | MR | Zbl

[13] Vulkov L. G., Zadorin A. I., “Two-Grid Algorithms for the Solution of 2D Semilinear Singularly Perturbed Convection-Diffusion Equations Using an Exponential Finite Difference Scheme”, AIP Conf. Proc., 1186:1 (2009), 371–379 | DOI | MR

[14] Zadorin A. I., Zadorin N. A., “Interpolyatsiya funktsii s pogransloinymi sostavlyayuschimi i ee primenenie v dvukhsetochnom metode”, Sib. elektr. matem. izv., 8 (2011), 247–267 | MR