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

This article is cited in 13 scientific papers (total in 13 papers)

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
Received: 01.09.1998

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
\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
\jour Siberian Math. J.
\yr 2001
\vol 42
\issue 3
\pages 568--578

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    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Vinogradova P.V., Zarubin A.G., “Projection–difference method for a linear operator–differential equation”, Differential Equations, 43:9 (2007), 1262–1270  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi
    4. Vinogradova P.V., “Error estimates for a projection–difference method for a linear differential–operator equation”, Differential Equations, 44:7 (2008), 970–979  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. P. V. Vinogradova, “Ob odnom chislennom metode resheniya zadachi Koshi dlya differentsialno-operatornogo uravneniya”, Sib. zhurn. industr. matem., 13:1 (2010), 34–45  mathnet  mathscinet
    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  mathnet  crossref  mathscinet  elib
    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  mathnet  crossref  elib  elib
    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  mathnet  crossref
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  zmath  adsnasa  isi  elib  elib
    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  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  elib  scopus
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