Geodesic
A two-grid method for a~nonlinear singular perturbation boundary value problem on the Shishkin scheme
Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 1, pp. 42-55.

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Unders consideration is some boundary value problem for a second order nonlinear singular perturbation ordinary differential equationd. An upwind scheme on the Shishkin mesh is applied. We use the Newton and Picard methods to resolve the difference schemeare investigated. To decrease number of arithmetical operations we use the two-grid method. Application of the Richardson extrapolation is shown to give almost second order accuracy of the difference scheme. The results of some numerical experiments are discussed.
Keywords: nonlinear differential equation, Shishkin mesh, difference scheme, iterative method, two-grid method, Richardson extrapolation.
Mots-clés : singular perturbation 
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A. I. Zadorin; S. V. Tikhovskaya. A two-grid method for a~nonlinear singular perturbation boundary value problem on the Shishkin scheme. Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 1, pp. 42-55. http://geodesic.mathdoc.fr/item/SJIM_2013_16_1_a4/

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

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

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

[4] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Sci., Singapore, 1996 | MR

[5] Fedorenko R. P., Vvedenie v vychislitelnuyu fiziku, Izd-vo MFTI, M., 1994

[6] Shaidurov V. V., Mnogosetochnye metody konechnykh elementov, Nauka, M., 1989 | MR

[7] Xu J., “A novel two-grid method for semilinear elliptic equation”, SIAM J. Sci. Comput., 15 (1994), 231–237 | DOI | MR | Zbl

[8] Axelsson O., Layton W., “A two-level discretization of nonlinear boundary value problems”, SIAM J. Numer. Anal., 33 (1996), 2359–2374 | DOI | MR | Zbl

[9] Vulkov L. G., Zadorin A. I., “Two-grid algorithms for an ordinary second order equation with exponential boundary layer in the solution”, Internat. J. Numer. Analysis and Modeling, 7:3 (2010), 580–592 | MR | Zbl

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

[11] Marchuk G. I., Shaidurov V. V., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979 | MR

[12] Shishkin G. I., Shishkina L. P., Difference Methods for Singular Perturbation Problems, Monographs and Surveys in Pure and Applied Mathematics, 140, Chapman Hall, Boca Raton, 2009 | MR | Zbl

[13] Ilin V. P., Metody konechnykh raznostei i konechnykh ob'emov dlya ellipticheskikh uravnenii, Izd-vo IVMiMG, Novosibirsk, 2001

[14] Shishkin G. I., “Metod povyshennoi tochnosti dlya kvazilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Sib. zhurn. vychisl. matematiki, 9:1 (2006), 81–108 | Zbl

[15] Roos H., Stynes M., Tobiska L., Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, 24, Springer-Verl., Berlin, 2008 | MR | Zbl

[16] Zadorin A. I., “Metod interpolyatsii na sguschayuscheisya setke dlya funktsii s pogransloinoi sostavlyayuschei”, Zhurn. vychisl. matematiki i mat. fiziki, 48:9 (2008), 1673–1684 | MR

[17] Natividad M. C., Stynes M., “Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh”, Appl. Numer. Math., 45:2 (2003), 315–329 | DOI | MR | Zbl