Skip to main content
SpringerLink
Log in

Solving a variational parabolic equation with the periodic condition by a projection-difference method with the Crank–Nicolson scheme in time

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

A solution to a smoothly solvable linear variational parabolic equation with the periodic condition is sought in a separable Hilbert space by an approximate projection-difference method using an arbitrary finite-dimensional subspace in space variables and the Crank–Nicolson scheme in time. Solvability, uniqueness, and effective error estimates for approximate solutions are proven. We establish the convergence of approximate solutions to a solution as well as the convergence rate sharp in space variables and time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin J.-P., Approximation of Elliptic Boundary-Value Problems, Dover Publ., Mineola (2007).

    MATH  Google Scholar 

  2. Lions J.-L. and Magenes E., Inhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  3. Bondarev A. S. and Smagin V. V., “Convergence of the projection-difference method for an approximate solution of a parabolic equation with periodic conditions for solutions,” Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., no. 2, 81–94 (2014).

    MATH  Google Scholar 

  4. Bondarev A. S., “Solvability of a variational parabolic equation with periodic conditions for solutions,” Vestn. Voronezh. Gos. Univ. Ser. Fiz. Mat., no. 4, 78–88 (2015).

    MATH  Google Scholar 

  5. Vaĭnikko G. M. and Oya P. È., “On convergence and the rate of convergence of the Galerkin method for abstract evolution equations,” Differ. Uravn., vol. 11, no. 7, 1269–1277 (1975).

    MathSciNet  Google Scholar 

  6. Smagin V. V., “Projection-difference method with the Crank–Nicolson scheme in time for the approximate solution of a parabolic equation with an integral condition for the solution,” Differ. Equ., vol. 51, no. 1, 116–126 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. Lions J.-L., Optimal Control of Systems Governed by Partial Differential Equations [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  8. Marchuk G. I. and Agoshkov V. I., An Introduction to Projection-Grid Methods [Russian], Nauka, Moscow (1981).

    MATH  Google Scholar 

  9. Smagin V. V., “Projection-difference methods for the approximate solution of parabolic equations with nonsymmetric operators,” Differ. Equ., vol. 37, no. 1, 128–137 (2001).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    A. S. Bondarev & V. V. Smagin

Authors
  1. A. S. Bondarev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Smagin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. S. Bondarev.

Additional information

Voronezh. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 4, pp. 761–771, July–August, 2017

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bondarev, A.S., Smagin, V.V. Solving a variational parabolic equation with the periodic condition by a projection-difference method with the Crank–Nicolson scheme in time. Sib Math J 58, 591–599 (2017). https://doi.org/10.1134/S0037446617040048

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446617040048

Keywords

Search

Navigation