Geodesic
Accuracy of difference schemes for nonlinear elliptic equations with non-restricted nonlinearity
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 3, pp. 41-52.

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We consider the first boundary-value problem for nonlinear elliptic equations with mixed derivatives and unrestricted nonlinearity. Difference scheme for solution of a given problem class and an iterative process implementing the scheme are constructed and studied. Rigorous study of the iterative process’ convergence is conducted. Existence and uniqueness of solution of nonlinear difference scheme approximating the original differential problem are proved. Estimates of convergence rates for difference schemes approximating nonlinear equation with non-restricted nonlinearity are obtained. These estimates are consistent with the smoothness of the sought solution.
Keywords: nonlinear elliptic equations, difference method of solving, accuracy of difference approximations, iterative process. 
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F. V. Lubyshev; M. È. Fairuzov; A. R. Manapova. Accuracy of difference schemes for nonlinear elliptic equations with non-restricted nonlinearity. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 3, pp. 41-52. http://geodesic.mathdoc.fr/item/SVMO_2017_19_3_a3/

[1] A. A. Samarskij, Teoriya raznostnyh skhem, Nauka, M., 1989 (In Russ.) | MR

[2] A. A. Samarskij, P. N. Vabishchevich, Vychislitel'naya teploperedacha, Knizhnyj dom «LIBROKOM», M., 2009 (In Russ.)

[3] A. A. Samarskij, R. D. Lazarov, V. L. Makarov, Raznostnye skhemy dlya differencial'nyh uravnenij s obobshchennymi resheniyami, Vysshaya shkola, M., 1987 (In Russ.)

[4] A. A. Samarskij, “Issledovanie tochnosti raznostnyh skhem dlya zadach s obobshchennymi resheniyami”, Aktual'nye problemy matematicheskoj fiziki i vychislitel'noj matematiki, 1984, 174-183 (In Russ.)

[5] P. P. Matus, “Dvuhslojnye raznostnye skhemy s peremennymi vesami”, Vestnik AN Belarusi. Seriya fiz.-mat. nauk, 4 (1993), 15-21 (In Russ.) | Zbl

[6] V. N. Abrashin, “Raznostnye skhemy dlya nelinejnyh giperbolicheskih uravnenij”, Differencial'nye uravneniya, 11:2 (1975), 294-308 (In Russ.) | MR | Zbl

[7] V. N. Abrashin, V. A. Asmolik, “Lokal'no-odnomernye raznostnye skhemy dlya mnogomernyh kvazilinejnyh giperbolicheskih uravnenij”, Differencial'nye uravneniya, 18:7 (1982), 1107-1117 (In Russ.) | MR

[8] P. P. Matus, “O bezuslovnoj skhodimosti nekotoryh raznostnyh skhem zadach gazovoj dinamiki”, Differencial'nye uravneniya, 21:7 (1985), 1227-1238 (In Russ.) | MR

[9] P. P. Matus, L. V. Stanishevskaya, “O bezuslovnoj skhodimosti raznostnyh skhem dlya nestacionarnyh kvazilinejnyh uravnenij matematicheskoj fiziki”, Differencial'nye uravneniya, 27:7 (1991), 1203-1219 (In Russ.) | MR

[10] P. P. Matus, M. N. Moskal'kov, V. S. Shcheglik, “Soglasovannye ocenki tochnosti metoda setok dlya nelinejnogo uravneniya vtorogo poryadka s obobshchennymi resheniyami”, Differencial'nye uravneniya, 31:7 (1995), 1249-1256 (In Russ.) | MR | Zbl

[11] V. S. Shcheglik, “Analiz raznostnoj skhemy, approksimiruyushchej tret'yu kraevuyu zadachu dlya nelinejnogo differencial'nogo uravneniya vtorogo poryadka”, Zhurnal vychisl. matem. i matem. fiziki, 37:8 (1997), 951-957 (In Russ.) | MR

[12] B. S. Jovanovic, E. Suli, Analysis of Finite Difference Schemes, Springer Series in Computational Mathematics, 46, Springer Science Business Media, London, 2014, 408 pp. | DOI | MR | Zbl

[13] B. S. Jovanovic, “Finite difference scheme for partial differential equations with weak solutions and irregular coefficients”, Comput. Methods Appl. Math., 4:1 (2004), 48-65 | MR | Zbl

[14] B. S. Jovanovic, P. P. Matus, V. S. Shchehlik, “The estimates of accuracy of difference schemes for the nonlinear heat equation with weak solutions”, Mathematical Modelling and Analysis (MMA), 5:1 (2000), 86-96 | DOI | MR | Zbl

[15] P. P. Matus, “O korrektnosti raznostnyh skhem dlya polulinejnogo uravneniya s obobshchennymi resheniyami”, ZHurnal vychisl. matem. i matem. fiziki, 50:12 (2010), 2155-2175 (In Russ.) | MR | Zbl

[16] P. Matus, “On convergence of difference schemes for IBVP for quasilinear parabolic equations with generalized solutions”, Comput. Methods Appl. Math., 14:3 (2014), 361-371 | DOI | MR | Zbl

[17] F. V. Lubyshev, Raznostnye approksimacii zadach optimal'nogo upravleniya sistemami, opisyvaemymi uravneniyami v chastnyh proizvodnyh, BashGU, Ufa, 1999 (In Russ.)

[18] F. V. Lubyshev, M. EH. Fajruzov, “Approksimacii zadach optimal'nogo upravleniya dlya polulinejnyh ehllipticheskih uravnenij s razryvnymi koehfficientami i sostoyaniyami, s upravleniyami v koehfficientah pri starshih proizvodnyh”, ZHurnal vychisl. matem. i matem. fiziki, 56:7 (2016), 1267-1293 (In Russ.) | DOI | Zbl

[19] S. L. Sobolev, Nekotorye primeneniya funkcional'nogo analiza v matematicheskoj fizike, Izd-vo SO AN SSSR, Novosibirsk, 1962 (In Russ.) | MR

[20] S. G. Mihlin, Linejnye uravneniya v chastnyh proizvodnyh, Vysshaya shkola, M., 1973 (In Russ.)