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 Sibirsk. Mat. Zh., 2001, Volume 42, Number 3, Pages 670–682 (Mi smj1451)

Energy error estimates for the projection-difference method with the Crank–Nicolson scheme for parabolic equations

V. V. Smagin

Voronezh State University

Abstract: We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.

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English version:
Siberian Mathematical Journal, 2001, 42:3, 568–578

Bibliographic databases:

UDC: 517.988.8

Citation: V. V. Smagin, “Energy error estimates for the projection-difference method with the Crank–Nicolson scheme for parabolic equations”, Sibirsk. Mat. Zh., 42:3 (2001), 670–682; Siberian Math. J., 42:3 (2001), 568–578

Citation in format AMSBIB
\Bibitem{Sma01} \by V.~V.~Smagin \paper Energy error estimates for the projection-difference method with the Crank--Nicolson scheme for parabolic equations \jour Sibirsk. Mat. Zh. \yr 2001 \vol 42 \issue 3 \pages 670--682 \mathnet{http://mi.mathnet.ru/smj1451} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1852243} \zmath{https://zbmath.org/?q=an:1066.65103} \elib{http://elibrary.ru/item.asp?id=804117} \transl \jour Siberian Math. J. \yr 2001 \vol 42 \issue 3 \pages 568--578 \crossref{https://doi.org/10.1023/A:1010483428596} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169277100014} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Smagin, “Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank–Nicolson Scheme”, Math. Notes, 74:6 (2003), 864–873
2. Vinogradova P.V., Zarubin A.G., “Projection–difference method for a linear operator–differential equation”, Differential Equations, 43:9 (2007), 1262–1270
3. Ashyralyev A., “Well–posedness of the modified Crank–Nicholson difference schemes in Bochner spaces”, Discrete and Continuous Dynamical Systems–Series B, 7:1 (2007), 29–51
4. Vinogradova P.V., “Error estimates for a projection–difference method for a linear differential–operator equation”, Differential Equations, 44:7 (2008), 970–979
5. Vinogradova P., “Convergence estimates of a projection–difference method for an operator–differential equation”, Journal of Computational and Applied Mathematics, 231:1 (2009), 1–10
6. P. V. Vinogradova, “Ob odnom chislennom metode resheniya zadachi Koshi dlya differentsialno-operatornogo uravneniya”, Sib. zhurn. industr. matem., 13:1 (2010), 34–45
7. P. V. Vinogradova, “Error estimates for projection-difference methods for differential equations with differentiable operators”, Russian Math. (Iz. VUZ), 54:7 (2010), 1–11
8. S. E. Zhelezovskii, “Error estimates in the projection-difference method for a hyperbolic-parabolic system of abstract differential equations”, Num. Anal. Appl., 3:3 (2010), 218–230
9. P. V. Vinogradova, A. G. Zarubin, “Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator”, Num. Anal. Appl., 3:4 (2010), 317–328
10. Zhelezovskii S.E., “Estimates for the accuracy of the projection-difference method for an abstract hyperbolic-parabolic system like systems of thermoelasticity equations”, Differ Equ, 46:7 (2010), 998–1010
11. M. I. Ivanov, I. A. Kremer, M. V. Urev, “Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium”, Comput. Math. Math. Phys., 52:3 (2012), 476–488
12. Ashyralyev A., “Well-Posedness of the Modified Crank-Nicholson Difference Schemes in C-Tau(Beta,Gamma)(E) and (C)Over-Tilde(Tau)(Beta,Gamma)(E) Spaces”, Appl. Math. Inf. Sci., 6:3 (2012), 543–554
13. Zhelezovskii S.E., “Error Estimate for a Symmetric Scheme of the Projection-Difference Method for an Abstract Hyperbolic-Parabolic System of the Type of Systems of Thermoelasticity Equations”, Differ. Equ., 48:7 (2012), 950–964