Cybernetics and Systems Analysis

, Volume 50, Issue 2, pp 278–287 | Cite as

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

  • S. V. Siryk


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.


finite element method Petrov–Galerkin method convection–diffusion equation logarithmic norm 


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  1. 1.
    K. Fletcher, Numerical Methods Based on the Galerkin Method [Russian translation], Mir, Moscow (1988).Google Scholar
  2. 2.
    H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin–Heidelberg (2008).MATHGoogle Scholar
  3. 3.
    C. Grossmann, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations, Springer, Berlin–Heidelberg (2007).CrossRefMATHGoogle Scholar
  4. 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. 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).CrossRefMATHMathSciNetGoogle Scholar
  6. 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).CrossRefMATHMathSciNetGoogle Scholar
  7. 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).CrossRefMATHMathSciNetGoogle Scholar
  8. 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).CrossRefMATHMathSciNetGoogle Scholar
  9. 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).CrossRefMathSciNetGoogle Scholar
  10. 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).Google Scholar
  11. 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).Google Scholar
  12. 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).CrossRefMATHMathSciNetGoogle Scholar
  13. 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.Google Scholar
  14. 14.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], 6th Edition, Nauka, Moscow (1999).Google Scholar
  15. 15.
    V. G. Mazja, Sobolev Spaces [in Russian], Izd-vo LGU, Leningrad (1985).CrossRefGoogle Scholar
  16. 16.
    S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton (N.J.) (1965).MATHGoogle Scholar
  17. 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. 18.
    J. Xu and L Zikatanov, “Some observations on Babuska and Brezzi theories,” BIT Num. Math., 94, 195–202 (2003).CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    R. Horn and C. Johnson, Matrix Analysis [Russian translation], Mir, Moscow (1989).Google Scholar
  20. 20.
    K. Dekker and Ya. Verver, Stability of Runge–Kutta Methods for Rigid Nonlinear Differential Equations [Russian translation], Mir, Moscow (1988).Google Scholar
  21. 21.
    G. Soderlind, “The logarithmic norm: History and modern theory,” BIT Num. Math., 46, 631–652 (2006).CrossRefMathSciNetGoogle Scholar
  22. 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).Google Scholar
  23. 23.
    A. A. Samarskii and A. V. Gulin, Numerical Methods of Mathematical Physics [in Russian], 2nd Edition, Nauchnyi Mir, Moscow (2003).Google Scholar
  24. 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. 25.
    A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia (1994).CrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

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

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