Skip to main content
SpringerLink
Log in

Accuracy and Stability of the Petrov–Galerkin Method for Solving the Stationary Convection-Diffusion Equation

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

The accuracy and stability of numerical solution of the stationary convection-diffusion equation by the finite element Petrov–Galerkin method are analyzed with the use of weight functions with different stabilization parameters as test functions, and estimates are obtained for the accuracy of the method depending on the choice of a collection of stabilization parameters. The convergence of the method is shown.

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. K. Fletcher, Numerical Methods Based on the Galerkin Method [Russian translation], Mir, Moscow (1988).

    Google Scholar 

  2. H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin–Heidelberg (2008).

    MATH  Google Scholar 

  3. C. Grossmann, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations, Springer, Berlin–Heidelberg (2007).

    Book  MATH  Google Scholar 

  4. T. P. Fries and H. G. Matthies, A Review of Petrov–Galerkin Stabilization Approaches and an Extension to Meshfree Methods, Informatik-Bericht Nr. 2004-01, Techn. Univ. of Braunschweig, Brunswick (2004).

    Google Scholar 

  5. T. J. R. Hughes, G. Scovazzi, and T. E. Tezduyar, “Stabilized methods for compressible flows,” J. Sci. Comput., 43, No. 3, 343–368 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  6. V. John and E. Schmeyer, “Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion,” Comput. Methods Appl. Mech. Eng., 198, 475–494 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  7. T. E. Tezduyar and M. Senga, “Stabilization and shock-capturing parameters in SUPG formulation of compressible flows,” Comput. Methods Appl. Mech. Eng., 195, 1621–1632 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Nadukandi, E. Onate, and J. Garcia, “A high-resolution Petrov–Galerkin method for the 1D convection–diffusion–reaction problem,” Comput. Methods Appl. Mech. Eng., 199 (9–12), 525–546 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Nadukandi, E. Onate, and J. Garcia, “A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part 2: A multidimensional extension,” Comput. Methods Appl. Mech. Eng., 213–216, 327–352 (2012).

    Article  MathSciNet  Google Scholar 

  10. N. N. Salnikov, S. V. Siryk, and I. A. Tereshchenko, “On the construction of a finite-dimensional mathematical model of convection–diffusion process with the use of the Petrov–Galerkin method,” Probl. Upravl. Inf., No. 3, 94–109 (2010).

  11. S. V. Siryk and N. N. Salnikov, “Numerical integration of the Burgers equation by the Petrov–Galerkin method with adaptive weight functions,” Probl. Upravl. Inf., No. 1, 94–110 (2012).

  12. D. F. Griffiths and J. Lorenz, “An analysis of the Petrov–Galerkin finite element method,” Comput. Methods Appl. Mech. Eng., 14, 39–64 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  13. K. W. Morton, “Finite element methods for non-self-adjoint problems,” in: P. R. Turner (ed.), Proc. SERC Summer School (Lancaster, 1981), Lect. Notes Math., 965, Springer, Berlin (1982), pp. 113–148.

  14. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], 6th Edition, Nauka, Moscow (1999).

    Google Scholar 

  15. V. G. Mazja, Sobolev Spaces [in Russian], Izd-vo LGU, Leningrad (1985).

    Book  Google Scholar 

  16. S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton (N.J.) (1965).

    MATH  Google Scholar 

  17. I. Babuska and A. K. Aziz, “Survey lectures on the mathematical foundations of the finite element method,” in: A. K. Aziz (ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Acad. Press, New York (1972), pp. 2–363.

    Google Scholar 

  18. J. Xu and L Zikatanov, “Some observations on Babuska and Brezzi theories,” BIT Num. Math., 94, 195–202 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Horn and C. Johnson, Matrix Analysis [Russian translation], Mir, Moscow (1989).

    Google Scholar 

  20. K. Dekker and Ya. Verver, Stability of Runge–Kutta Methods for Rigid Nonlinear Differential Equations [Russian translation], Mir, Moscow (1988).

    Google Scholar 

  21. G. Soderlind, “The logarithmic norm: History and modern theory,” BIT Num. Math., 46, 631–652 (2006).

    Article  MathSciNet  Google Scholar 

  22. A. I. Perov, “Sufficient conditions of stability of linear systems with constant coefficients in critical cases. I,” Automatics and Telemechanics, No. 12, 80–89 (1997).

  23. A. A. Samarskii and A. V. Gulin, Numerical Methods of Mathematical Physics [in Russian], 2nd Edition, Nauchnyi Mir, Moscow (2003).

    Google Scholar 

  24. F. R. Gantmakher and M. G. Krein, Oscillation Matrices and Kernels and Small Oscillations of Mechanical Systems [in Russian], 2nd Edition, GITTL, Moscow (1950).

    Google Scholar 

  25. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia (1994).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine

    S. V. Siryk

Authors
  1. S. V. Siryk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Siryk.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 132–143, March–April, 2014.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Siryk, S.V. Accuracy and Stability of the Petrov–Galerkin Method for Solving the Stationary Convection-Diffusion Equation. Cybern Syst Anal 50, 278–287 (2014). https://doi.org/10.1007/s10559-014-9615-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-014-9615-7

Keywords

Search

Navigation