Geodesic
Error estimates for projection-difference methods for differential equations with differentiable operators
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2010), pp. 3-15.

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We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator $A(t)$ and a subordinate linear operator $K(t)$, whose definition domain is independent of $t$. Operators $A(t)$ and $K(t)$ are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to $A(0)$.
Keywords: Hilbert space, Cauchy problem, Galyorkin method, three-level scheme, operator equation, orthoprojector
Mots-clés : convergence rate. 
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P. V. Vinogradova. Error estimates for projection-difference methods for differential equations with differentiable operators. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2010), pp. 3-15. http://geodesic.mathdoc.fr/item/IVM_2010_7_a0/

[1] Oya P. E., “O skhodimosti i ustoichivosti metoda Galërkina dlya parabolicheskikh uravnenii s differentsiruemymi operatorami”, Uchen. zap. Tartusk. un-ta, 374 (1975), 194–210

[2] Vainikko G. M., Oya P. E., “O skhodimosti i bystrote skhodimosti metoda Galërkina dlya abstraktnykh evolyutsionnykh uravnenii”, Differents. uravneniya, 11:7 (1975), 1269–1277 | MR | Zbl

[3] Krein S. G., Lineinye differentsialnye uravneniya v banakhovykh prostranstvakh, Nauka, M., 1967 | MR

[4] Lions Zh., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR | Zbl

[5] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[6] Fattorini H. O., The Cauchy problem, Addison-Wesley Publishing Company, Massachusets, 1983 | MR

[7] Sobolevskii P. E., “Ob uravneniyakh s operatorami, obrazuyuschimi ostryi ugol”, DAN SSSR, 116:5 (1957), 754–757 | MR

[8] Smagin V. V., “Otsenki skorosti skhodimosti proektsionnogo i proektsionno-raznostnogo metodov dlya slabo razreshimykh parabolicheskikh uravnenii”, Matem. sb., 188:3 (1997), 143–160 | MR | Zbl

[9] Zarubin A. G., “O skorosti skhodimosti metoda Faedo–Galërkina dlya lineinykh nestatsionarnykh uravnenii”, Differents. uravneniya, 18:4 (1982), 639–645 | MR | Zbl

[10] Zarubin A. G., “O skorosti skhodimosti metoda Faedo–Galërkina dlya kvazilineinykh nestatsionarnykh operatornykh uravnenii”, Differents. uravneniya, 26:12 (1990), 2051–2059 | MR | Zbl

[11] Hausenblas E., “Numerical analysis of semilinear stochastic evolution equations in Banach spaces”, J. Comput. Appl. Math., 147:2 (2002), 485–516 | DOI | MR | Zbl

[12] Lyashko A. D., “Proektsionno-raznostnye skhemy dlya giperbolicheskikh uravnenii s vyrozhdayuschimsya ellipticheskim operatorom vysokogo poryadka”, Differents. uravneniya, 43:7 (2007), 972–977 | MR | Zbl

[13] Li H., Liu R., “The space-time finite element method for parabolic problems”, Appl. Mathematics and Mechanics, 22:6 (2001), 687–700 | DOI | MR | Zbl

[14] Turetaev I. D., “Tochnye otsenki gradienta pogreshnosti proektsionno-raznostnykh skhem dlya parabolicheskikh uravnenii v proizvolnoi oblasti”, Zhurn. vychisl. matem. i matem. fiz., 26:11 (1986), 1748–1751 | MR

[15] Smagin V. V., “Energeticheskie otsenki pogreshnosti proektsionno-raznostnogo metoda so skhemoi Kranka–Nikolson dlya parabolicheskogo uravneniya”, Sib. matem. zhurn., 42:3 (2001), 670–682 | MR | Zbl

[16] Ashyralyev A., Sobolevskii P. E., “Well-posed solvability of the Cauchy problem for difference equation of parabolic type”, Nonlinear Anal., 24:2 (1995), 257–264 | DOI | MR | Zbl

[17] Ashyralyev A., Piskarev S., Weis L., “On well-posedness of difference schemes for abstract parabolic equations”, Numer. Funct. Anal. and Optimization, 23:7–8 (2002), 669–693 | DOI | MR | Zbl

[18] Mikhlin S. G., Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970 | MR | Zbl

[19] Zarubin A. G., Tiunchik M. F., “O metode Rote–Galërkina dlya odnogo klassa lineinykh nestatsionarnykh uravnenii”, Differents. uravneniya, 19:12 (1983), 2141–2148 | MR | Zbl

[20] Zarubin A. G., “K metodu Rote i Rote–Galërkina dlya kvazilineinogo differentsialno-operatornogo uravneniya”, Differents. uravneniya, 32:12 (1996), 1683–1690 | MR | Zbl

[21] Vinogradova P. V., “Otsenki pogreshnosti proektsionno-raznostnogo metoda dlya lineinogo differentsialno-operatornogo uravneniya”, Differents. uravneniya, 44:7 (2008), 942–951 | MR | Zbl

[22] Demidovich V. B., “Ob asimptoticheskom povedenii reshenii konechno-raznostnykh uravnenii”, Differents. uravneniya, 10:12 (1974), 2267–2278 | Zbl

[23] Zarubin A. G., Issledovanie metodom drobnykh stepenei proektsionnykh protsedur resheniya lineinykh i kvazilineinykh uravnenii, Diss. $\dots$ dokt. fiz.-matem. nauk, Khabarovsk, 1986

[24] Samarskii A. A., Gulin A. V., Chislennye metody, Nauka, M., 1989 | MR

[25] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967