Geodesic
Error Estimates for Schemes of the Projection-Difference Method
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 38-49.

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We study the convergence of the three-layer scheme of the projection-difference method for abstract quasilinear hyperbolic equations in Hilbert space. We establish asymptotic energy error estimates for an arbitrary choice of finite-dimensional subspaces in which the approximation problems are solved.
Keywords: projection-difference method, three-layer scheme, quasilinear hyperbolic equations, asymptotic error estimates, energy norm, uniform norm. 
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S. E. Zhelezovsky. Error Estimates for Schemes of the Projection-Difference Method. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 38-49. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a5/

[1] T. Dupont, “$L^2$-estimates for Galerkin methods for second order hyperbolic equations”, SIAM J. Numer. Anal., 10:5 (1973), 880–889 | DOI | MR | Zbl

[2] G. A. Baker, “Error estimates for finite element methods for second order hyperbolic equations”, SIAM J. Numer. Anal., 13:4 (1976), 564–576 | DOI | MR | Zbl

[3] T. Geveci, “On the convergence of Galerkin approximation schemes for second order hyperbolic equations in energy and negative norms”, Math. Comput., 42:166 (1984), 393–415 | DOI | MR | Zbl

[4] A. A. Zlotnik, “Otsenki skorosti skhodimosti proektsionno-setochnykh metodov dlya giperbolicheskikh uravnenii vtorogo poryadka”, Vychislitelnye protsessy i sistemy, 8, Nauka, M., 1991, 116–167 | MR

[5] S. E. Zhelezovskii, “Otsenki skorosti skhodimosti proektsionno-raznostnogo metoda dlya giperbolicheskikh uravnenii”, Izv. vuzov. Matem., 2002, no. 1, 21–30 | MR | Zbl

[6] J. E. Dendy, Jr., “An analysis of some Galerkin schemes for the solution of nonlinear time-dependent problems”, SIAM J. Numer. Anal., 12:4 (1975), 541–565 | DOI | MR | Zbl

[7] G. A. Baker, V. A. Dougalis, O. Karakashian, “On multistep-Galerkin discretizations of semilinear hyperbolic and parabolic equations”, Nonlinear Anal., Theory, Meth. and Appl., 4:3 (1980), 579–597 | DOI | MR | Zbl

[8] S. E. Zhelezovskii, “Otsenki skorosti skhodimosti metoda Galerkina dlya abstraktnogo giperbolicheskogo uravneniya”, Matem. zametki, 69:2 (2001), 223–234 | MR | Zbl

[9] S. E. Zhelezovskii, A. D. Lyashko, “Otsenki pogreshnosti metoda Galerkina dlya kvazilineinykh giperbolicheskikh uravnenii”, Differents. uravneniya, 37:7 (2001), 941–949 | MR | Zbl

[10] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[11] S. G. Krein, Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967 | MR

[12] N. F. Morozov, “Issledovanie kolebanii prizmaticheskogo sterzhnya pod deistviem poperechnoi nagruzki”, Izv. vuzov. Matem., 1965, no. 3, 121–125 | Zbl

[13] A. A. Samarskii, Teoriya raznostnykh skhem, 3-e izd., Nauka, M., 1989 | MR

[14] A. A. Samarskii, A. V. Gulin, Ustoichivost raznostnykh skhem, Nauka, M., 1973 | Zbl

[15] I. I. Vorovich, “O nekotorykh pryamykh metodakh v nelineinoi teorii kolebanii pologikh obolochek”, Izv. AN SSSR. Ser. matem., 21:6 (1957), 747–784

[16] S. E. Zhelezovskii, “O suschestvovanii i edinstvennosti resheniya i o skorosti skhodimosti metoda Bubnova–Galerkina dlya odnoi kvazilineinoi evolyutsionnoi zadachi v gilbertovom prostranstve”, Izv. vuzov. Matem., 1998, no. 10, 37–45 | MR | Zbl

[17] E. Bekkenbakh, R. Bellman, Neravenstva, Mir, M., 1965 | MR

[18] V. B. Demidovich, “Ob odnom priznake ustoichivosti raznostnykh uravnenii”, Differents. uravneniya, 5:7 (1969), 1247–1255 | MR | Zbl

[19] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 5-e izd., Nauka, M., 1981 | MR

[20] F. Syarle, Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[21] O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[22] S. G. Mikhlin, “Po povodu metoda Rittsa”, Dokl. AN SSSR, 106:3 (1956), 391–394