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Stabilized finite-element method for the stationary Navier-Stokes equations

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Abstract

A stabilized finite-element method for the two-dimensional stationary incompressible Navier-Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation of the stationary Navier-Stokes equations. By satisfying this condition, the stability of the Q1P0 quadrilateral element and the P1P0 triangular element are established. Moreover, the well-posedness and the optimal error estimate of the stabilized finite-element method for the stationary Navier-Stokes equations are obtained. Finally, some numerical tests to confirm the theoretical results of the stabilized finite-element method are provided.

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References

  1. V. Girault P.A. Raviart (1981) Finite Element Method for Navier-Stokes Equations: Theory and Algorithms Springer-Verlag Berlin Heidelberg 202

    Google Scholar 

  2. J. Boland R.A. Nicolaides (1983) ArticleTitleStability of finite elements under divergence constraints SIAM J. Numer. Anal. 20 722–731 Occurrence Handle10.1137/0720048

    Article  Google Scholar 

  3. R. Stenberg (1984) ArticleTitleAnalysis of mixed finite elements for the Stokes problem: A unified approach Math. Comp. 42 9–23

    Google Scholar 

  4. N. Kechkar D. Silvester (1992) ArticleTitleAnalysis of locally stabilized mixed finite element methods for the Stokes problem Math. Comp. 58 1–10

    Google Scholar 

  5. D.J. Silvester N. Kechkar (1990) ArticleTitleStabilised bilinear-constant velocity–pressure finite elements for the conjugate gradient solution of the Stokes problem. Comp Methods Appl. Mech. Engng 79 71–86 Occurrence Handle10.1016/0045-7825(90)90095-4

    Article  Google Scholar 

  6. D. Kay D. Silvester (2000) ArticleTitleA posteriori error estimation for stabilized mixed approximations of the Stokes equations SIAM J. Sci. Comp. 21 1321–1337 Occurrence Handle10.1137/S1064827598333715

    Article  Google Scholar 

  7. S. Norburn D. Silvester (1998) ArticleTitleStabilised vs. stable mixed methods for incompressible flow Comput. Methods Appl. Mech. Engng. 166 1–10 Occurrence Handle10.1016/S0045-7825(98)00078-4

    Article  Google Scholar 

  8. D. Silvester A. Wathen (1994) ArticleTitleFast interative solution of stabilised Stokes systems, Part II: Using general block preconditioners SIAM J. Numer. Anal. 31 1352–1367 Occurrence Handle10.1137/0731070

    Article  Google Scholar 

  9. D. Braess (1997) Finite Elements, Theory, Fast Solvers and Applications in Solid Mechnics Cambridge University Press Cambridge 323

    Google Scholar 

  10. J. Pitkäranta T. Saarinen (1985) ArticleTitleA multigrid version of a simple finite element method for the Stokes problem Math. Comput. 45 1–14

    Google Scholar 

  11. J.G. Heywood R. Rannacher (1982) ArticleTitleFinite element approximation of the nonstationary Navier-Stokes problem, I: Regularity of solutions and second-order error estimates for spatial discretization SIAM J. Numer. Anal. 19 275–311 Occurrence Handle10.1137/0719018

    Article  Google Scholar 

  12. R.B. Kellogg J.E. Osborn (1976) ArticleTitleA regularity result for the Stokes problem in a convex polygon J. Funct. Anal. 21 397–431 Occurrence Handle10.1016/0022-1236(76)90035-5

    Article  Google Scholar 

  13. A. AitOu Ammi M. Marion (1994) ArticleTitleNonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier-Stokes equations Numer. Math. 68 189–213 Occurrence Handle10.1007/s002110050056

    Article  Google Scholar 

  14. R. Temam (1984) Navier-Stokes Equations, Theory and Numerical Analysis North-Holland Amsterdam 526

    Google Scholar 

  15. P.G. Ciarlet (1978) The Finite Element Method for Elliptic Problems North-Holland Amsterdam 519

    Google Scholar 

  16. Sani R.L., Gresho P.M., Lee R.L. and D.F. Griffiths, The cause and cure(?) of the spuious pressures generated by certain finite element method solutions of the incompressible Navier-Stokes equations. Parts 1 and 2. Int. J. Numer. Methods Fluids 1 (1981) 17–43; 171–206.

    Google Scholar 

  17. T.J.R. Hughes L.P. Franca (1987) ArticleTitleA new finite element formulation for CFD: VIThe I Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Engng. 65 85–97

    Google Scholar 

  18. W. Layton L. Tobiska (1998) ArticleTitleA two-level method with backtraking for the Navier-Stokes equations SIAM J. Numer. Anal. 35 2035–2054 Occurrence Handle10.1137/S003614299630230X

    Article  Google Scholar 

  19. A.J. Baker (1983) Finite Element Computational Fluid Mechanics Hemisphere Pub. Corp Washington 510

    Google Scholar 

  20. T.J.R. Hughes W.K. Liu A. Brooks (1979) ArticleTitleReview of finite element analysis of incompressible viscous flow by the penalty function formulations J. Comp Phys. 30 1–60 Occurrence Handle10.1016/0021-9991(79)90086-X

    Article  Google Scholar 

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Correspondence to Yinnian He.

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He, Y., Wang, A. & Mei, L. Stabilized finite-element method for the stationary Navier-Stokes equations. J Eng Math 51, 367–380 (2005). https://doi.org/10.1007/s10665-004-3718-5

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  • DOI: https://doi.org/10.1007/s10665-004-3718-5

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