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A Single-Step Characteristic-Curve Finite Element Scheme of Second Order in Time for the Incompressible Navier-Stokes Equations

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Abstract

In this paper we present a new single-step characteristic-curve finite element scheme of second order in time for the nonstationary incompressible Navier-Stokes equations. After supplying correction terms in the variational formulation, we prove that the scheme is of second order in time. The convergence rate of the scheme is numerically recognized by computational results.

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References

  1. Baba, K., Tabata, M.: On a conservative upwind finite element scheme for convective diffusion equations. RAIRO Anal. 15, 3–25 (1981)

    MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  3. Bristeau, M.O., Glowinski, R., Mantel, B., Periaux, J., Perrier, P., Pironneau, O.: A finite element approximation of Navier-Stokes equations for incompressible viscous fluids. Iterative methods of solution. In: Rautmann, R. (ed.) Approximation Methods for Navier-Stokes Problems. Lecture Notes in Mathematics, vol. 771, pp. 78–128. Springer, Berlin (1980)

    Chapter  Google Scholar 

  4. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Boukir, K., Maday, Y., Metivet, 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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    MATH  Google Scholar 

  7. Douglas Jr., J., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)

    Article  MathSciNet  Google Scholar 

  8. Franca, L.P., Frey, S.L.: Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 99, 209–233 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. FreeFem, http://www.freefem.org/

  10. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)

    MATH  Google Scholar 

  11. Hughes, T.J.R., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Eng. 73, 173–189 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hughes, T.J.R., Tezduyar, T.E.: Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput. Methods Appl. Mech. Eng. 45, 217–284 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hansbo, P., Johnson, C.: Adaptive streamline diffusion methods for compressible flow using conservation variables. Comput. Methods Appl. Mech. Eng. 87, 267–280 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge Univ. Press, Cambridge (1987)

    MATH  Google Scholar 

  15. Le Beau, G.J., Ray, S.E., Aliabadi, S.K., Tezduyar, T.E.: SUPG finite element computation of compressible flows with the entropy and conservation variables formulations. Comput. Methods Appl. Mech. Eng. 104, 397–422 (1993)

    Article  MATH  Google Scholar 

  16. Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38, 309–332 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pironneau, O.: Finite Element Methods for Fluids. Wiley, New York (1989)

    Google Scholar 

  18. Pironneau, O., Liou, J., Tezduyar, T.: Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advection-diffusion equation with time-dependent domains. Comput. Methods Appl. Mech. Eng. 100, 117–141 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rui, H., Tabata, M.: A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92, 161–177 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)

    MATH  Google Scholar 

  21. Süli, E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53, 459–483 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  22. Tabata, M.: A finite element approximation corresponding to the upwind finite differencing. Mem. Numer. Math. 4, 47–63 (1977)

    MATH  MathSciNet  Google Scholar 

  23. Tabata, M.: Discrepancy between theory and real computation on the stability of some finite element schemes. J. Comput. Appl. Math. 199, 424–431 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tabata, M., Fujima, S.: Finite-element analysis of high Reynolds number flows past a circular cylinder. J. Comput. Appl. Math. 38, 411–424 (1991)

    Article  MATH  Google Scholar 

  25. Tabata, M., Fujima, S.: Robustness of a characteristic finite element scheme of second order in time increment. In Groth, C., Zingg, D.W. (eds.) Computational Fluid Dynamics 2004, pp. 177–182. Springer, Berlin (2006)

    Chapter  Google Scholar 

  26. Tabata, M., Tagami, D.: Error estimates for finite element approximations of drag and lift in nonstationary Navier-Stokes flows. Jpn J. Ind. Appl. Math. 17, 371–389 (2000)

    Article  MathSciNet  Google Scholar 

  27. Tezduyar, T.: Stabilized finite element formulations for incompressible flow computations. Adv. Appl. Mech. 28, 1–44 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Thomasset, F.: Implementation of Finite Element Methods for Navier-Stokes Equations. Springer, New York (1981)

    MATH  Google Scholar 

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Correspondence to Hirofumi Notsu.

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Notsu, H., Tabata, M. A Single-Step Characteristic-Curve Finite Element Scheme of Second Order in Time for the Incompressible Navier-Stokes Equations. J Sci Comput 38, 1–14 (2009). https://doi.org/10.1007/s10915-008-9217-5

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  • DOI: https://doi.org/10.1007/s10915-008-9217-5

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