Stabilized Lagrange–Galerkin Schemes of First- and Second-Order in Time for the Navier–Stokes Equations

Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

Two stabilized Lagrange–Galerkin schemes for the Navier–Stokes equations are reviewed. The schemes are based on a combination of the Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure, the numbers of degrees of freedom are much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams–Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme, whose theoretical result is also recognized numerically.

Notes

Acknowledgements

This work was supported by JSPS (the Japan Society for the Promotion of Science) under the Japanese–German Graduate Externship (Mathematical Fluid Dynamics) and Grant-in-Aid for Scientific Research (S), No. 24224004. The authors are indebted to JSPS also for Grant-in-Aid for Young Scientists (B), No. 26800091 to the first author and for Grant-in-Aid for Scientific Research (C), No. 25400212 to the second author.

References

  1. 1.
    Achdou, Y., Guermond, J.-L.: Convergence analysis of a finite element projection/Lagrange–Galerkin method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 37, 799–826 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994)CrossRefMATHGoogle Scholar
  3. 3.
    Boukir, K., Maday, Y., Métivet, B., Razafindrakoto, E.: A high-order characteristics/finite element method for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 25, 1421–1454 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Hackbusch, W. (ed.) Efficient Solutions of Elliptic Systems. Vieweg, Wiesbaden (1984)Google Scholar
  5. 5.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  6. 6.
    Ewing, R.E., Russell, T.F.: Multistep Galerkin methods along characteristics for convection-diffusion problems. In: Vichnevetsky, R., Stepleman, R.S. (eds.) Advances in Computer Methods for Partial Differential Equations. IMACS, vol. IV, pp. 28–36 (1981)Google Scholar
  7. 7.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  8. 8.
    Morton, K.W., Priestley, A., Süli, E.: Stability of the Lagrange–Galerkin method with non-exact integration. ESAIM: Math. Model. Numer. Anal. 22, 625–653 (1988)MATHGoogle Scholar
  9. 9.
    Notsu, H.: Numerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme. Trans. Jpn. Soc. Comput. Eng. Sci. 2008, 20080032 (2008)Google Scholar
  10. 10.
    Notsu, H., Tabata, M.: A combined finite element scheme with a pressure stabilization and a characteristic-curve method for the Navier–Stokes equations. Trans. Jpn. Soc. Ind. Appl. Math. 18, 427–445 (2008) (in Japanese)Google Scholar
  11. 11.
    Notsu, H., Tabata, M.: Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. J. Sci. Comput. 65, 940–955 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Notsu, H., Tabata, M.: Error estimates of a stabilized Lagrange–Galerkin scheme for the Navier–Stokes equations. ESAIM: Math. Model. Numer. Anal. 50, 361–380 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Notsu, H., Tabata, M.: Error estimates of a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. In: Shibata, Y., Suzuki, Y. (eds.) Mathematical Fluid Dynamics, Present and Future. Springer, Berlin (to appear)Google Scholar
  14. 14.
    Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math. 38, 309–332 (1982)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ravindran, S.S.: Convergence of extrapolated BDF2 finite element schemes for unsteady penetrative convection model. Numer. Funct. Anal. Optim. 33, 48–79 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rui, H., Tabata, M.: A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92, 161–177 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs, NJ (1971)MATHGoogle Scholar
  18. 18.
    Süli, E.: Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math. 53, 459–483 (1988)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tabata, M.: Discrepancy between theory and real computation on the stability of some finite element schemes. J. Comput. Appl. Math. 199, 424–431 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Tabata, M., Uchiumi, S.: A Lagrange–Galerkin scheme with a locally linearized velocity for the Navier–Stokes equations. arXiv:1505.06681 [math.NA] (2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsKanazawa UniversityKakuma, KanazawaJapan
  2. 2.Department of MathematicsWaseda UniversityTokyoJapan

Personalised recommendations