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Ukrainian Mathematical Journal

, Volume 67, Issue 7, pp 1062–1090 | Cite as

Estimation of the Accuracy of Finite-Element Petrov–Galerkin Method in Integrating the One-Dimensional Stationary Convection-Diffusion-Reaction Equation

  • S. V. Sirik
Article

The accuracy and convergence of the numerical solutions of a stationary one-dimensional linear convection-diffusion-reaction equation (with Dirichlet boundary conditions) by the Petrov–Galerkin finiteelement method with piecewise-linear basis functions and piecewise-quadratic weighting functions are analyzed and the accuracy estimates of the method are obtained in certain norms depending on the choice of the collection of stabilization parameters of weight functions.

Keywords

Weight Function Galerkin Method Convective Component Logarithmic Norm Explicit Analytic Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B. A. Finlayson, Numerical Methods for Problems with Moving Fronts, Ravenna Park Publishing, Seattle (1992).Google Scholar
  2. 2.
    V. S. Deineka, I. V. Sergienko, and V. V. Skopetskii, Mathematical Models and Methods of Analysis in Problems with Discontinuous Solutions [in Russian], Naukova Dumka, Kiev (1995).Google Scholar
  3. 3.
    H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin (2008).zbMATHGoogle Scholar
  4. 4.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972).Google Scholar
  5. 5.
    A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for the Solution of Convection-Diffusion Equations [in Russian], Nauka, Moscow (2009).Google Scholar
  6. 6.
    C. Grossmann, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations, Springer, Berlin (2007).CrossRefzbMATHGoogle Scholar
  7. 7.
    T. P. Fries and H. G. Matthies, A Review of Petrov–Galerkin Stabilization Approaches and an Extension to Mesh-Free Methods, Technische Universität Braunschweig, Informatikbericht 2004-1, Brunswick (2004).Google Scholar
  8. 8.
    T. J. R. Hughes, G. Scovazzi, and T. E. Tezduyar, “Stabilized methods for compressible flows,” J. Sci. Comput., 43, 343–368 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    V. John and E. Schmeyer, “Finite-element methods for time-dependent convection-diffusion-reaction equations with small diffusion,” Comput. Meth. Appl. Mech. Eng., 198, 475–494 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    S. V. Sirik, “Exactness and stability of the Petrov–Galerkin method in the integration of the stationary convection-diffusion equation,” Kibernet. Sistem. Anal., 50, No. 2, 32–143 (2014).Google Scholar
  11. 11.
    D. F. Griffiths and J. Lorenz, “An analysis of the Petrov–Galerkin finite element method,” Comput. Meth. Appl. Mech. Eng., 14, 39–64 (1978).CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    K. W. Morton, “Finite element methods for nonself-adjoint problems,” in: P. R. Turner (editor), Proc. of the SERC Summer School (Lancaster, 1981), 965, Springer, Berlin (1982), pp. 113–148.Google Scholar
  13. 13.
    D. F. Griffiths, “Discretized eigenvalue problems, LBB constants and stabilization,” Numer. Anal., Longman, Edinburgh, 57–75 (1996).Google Scholar
  14. 14.
    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1983).CrossRefzbMATHGoogle Scholar
  15. 15.
    S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton (1965).zbMATHGoogle Scholar
  16. 16.
    A. R. Mitchell and R. Wait, The Finite Element Method in Partial Differential Equations, Wiley, New York (1977).zbMATHGoogle Scholar
  17. 17.
    K. Rektorys, Variational Methods in Mathematics, Science and Engineering, Reidel, Dordrecht (1980).zbMATHGoogle Scholar
  18. 18.
    O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar
  19. 19.
    I. Babuška and A. K. Aziz, “Survey lectures on the mathematical foundations of the finite element method,” in: A. K. Aziz (editor), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York (1972), pp. 2–363.Google Scholar
  20. 20.
    N. N. Sal’nikov, S. V. Sirik, and I. A. Tereshchenko, “On the construction of a finite-dimensional mathematical model of the convection-diffusion process with the use of the Petrov–Galerkin method,” Probl. Upravl. Inform., No. 3, 94–109 (2010).Google Scholar
  21. 21.
    S. V. Sirik, N. N. Sal’nikov, and V. K. Beloshapkin, “The choice of weight functions in the Petrov–Galerkin method for the integration of linear one-dimensional convection-diffusion equations,” Upravl. Sist. Mash., No. 1, 38–47 (2014).Google Scholar
  22. 22.
    J. Xu and L. Zikatanov, “Some observations on the Babuška and Brezzi theories,” Numer. Math., 94, 195–202 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge (1986).Google Scholar
  24. 24.
    K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam (1984).zbMATHGoogle Scholar
  25. 25.
    C. A. Desoer and H. Haneda, “The measure of a matrix as a tool to analyze computer algorithms for circuit analysis,” IEEE Trans. Circ. Theory, 19, No. 5, 480–486 (1972).CrossRefMathSciNetGoogle Scholar
  26. 26.
    A. A. Samarskii and A. V. Gulin, Numerical Methods of Mathematical Physics [in Russian], Nauch. Mir, Moscow (2003).Google Scholar
  27. 27.
    S. Noschese, L. Pasquini, and L. Reichel, “Tridiagonal Toeplitz matrices: properties and novel applications,” Numer. Linear Algebra Appl., 20, No. 2, 302–326 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Berlin (2007).Google Scholar
  29. 29.
    V. C. L. Hutson and J. S. Pym, Application of Functional Analysis and Operator Theory, Academic Press, London (1983).Google Scholar
  30. 30.
    K.-J. Bathe, D. Hendriana, F. Brezzi, and G. Sangalli, “Inf-sup testing of upwind methods,” Int. J. Numer. Meth. Eng., 48, 745–760 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    G. Sangalli, “Numerical evaluation of finite element methods in convection-diffusion problems,” Calcolo, 37, 233–251 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    G. Sangalli, “Numerical evaluation of FEM with application to the 1-D advection-diffusion problem,” Math. Models Meth. Appl. Sci., 12, No. 2, 205–228. (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    A. Buffa, C. de Falco, and G. Sangalli, “Isogeometric analysis: stable elements for the 2D Stokes equation,” Int. J. Numer. Meth. Fluids, 65, 1407–1422 (2011).CrossRefzbMATHGoogle Scholar
  34. 34.
    E. Abramov, H. Kvasnytsya, and H. Shynkarenko, “Partially quadratic and cubic approximations of h -adaptive FEM for onedimensional boundary-value problems,” Visn. Lviv. Univ., Ser. Prykl. Mat. Inform., Issue 17, 47–61 (2011).Google Scholar
  35. 35.
    V. M. Trushevs’kyi and H. Shynkarenko, “Parallel approximation of elliptic boundary-value problems with artificial neuroboundary with radial basis functions,” Visn. Lviv. Univ., Ser. Prykl. Mat. Inform., Issue 22, 108–117 (2014).Google Scholar
  36. 36.
    A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia (1994).CrossRefzbMATHGoogle Scholar
  37. 37.
    C. A. J. Fletcher, Computational Galerkin Methods, Springer, New York (1984).CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. V. Sirik
    • 1
  1. 1.“Kiev Polytechnic Institute” Ukrainian National Technical UniversityKievUkraine

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