Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AN A POSTERIORI ERROR ESTIMATE FOR MIXED FINITE ELEMENT APPROXIMATIONS OF THE NAVIER-STOKES EQUATIONS

  • Elakkad, Abdeslam (Laboratoire Genie Mecanique Faculte des Sciences et Techniques) ;
  • Elkhalfi, Ahmed (Laboratoire Genie Mecanique Faculte des Sciences et Techniques) ;
  • Guessous, Najib (Departement de mathematiques et informatique Ecole normale Superieure de Fes)
  • Received : 2009.11.25
  • Accepted : 2010.08.04
  • Published : 2011.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this work, a numerical solution of the incompressible Navier-Stokes equations is proposed. The method suggested is based on an algorithm of discretization by mixed finite elements with a posteriori error estimation of the computed solutions. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

Keywords

References

  1. M. Ainsworth and J. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience [John Wiley & Sons], New York, 2000.
  2. M. Ainsworth and J. Oden, A posteriori error estimators for the Stokes and Oseen equations, SIAM J. Numer. Anal. 34 (1997), no. 1, 228-245. https://doi.org/10.1137/S0036142994264092
  3. R. Araya, A. H. Poza, and F. Valentin, On a hierarchical estimator driven by a stabilized method for the reactive incompressible Navier-Stokes equations, Preprint submitted to Elsevier, 2008.
  4. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283-301. https://doi.org/10.1090/S0025-5718-1985-0777265-X
  5. W. Bao and J. W. Barrett, A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-Newtonian flow, RAIRO Model. Math. Anal. Numer. 32 (1998), no. 7, 843-858. https://doi.org/10.1051/m2an/1998320708431
  6. E. Barragy and G. F. Carey, Stream function-vorticity driven cavity solution using p finite elements, Comput. Fluids 26 (1997), 453-468. https://doi.org/10.1016/S0045-7930(97)00004-2
  7. M. Benzi and J. Liu, An efficient solver for the incompressible Navier-Stokes equations in rotation form, SIAM J. Sci. Comput. 29 (2007), no. 5, 1959-1981. https://doi.org/10.1137/060658825
  8. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991.
  9. C. Carstensen and S. A. Funken, A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems, Math. Comp. 70 (2001), no. 236, 1353-1381.
  10. G. Chavent and J. Jaffre, Mathematical Models and Finite Elements for Reservoir Simulation, Elsevier Science Publishers B. V., Netherlands, 1986.
  11. P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of numerical analysis, Vol. II, 17-351, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991.
  12. H. C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput. 20 (1999), no. 4, 1299-1316. https://doi.org/10.1137/S1064827596312547
  13. H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics, Oxford University Press, New York, 2005.
  14. E. Erturk, T. C. Corke, and C. Gokcol, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids 48 (2005), 747-774. https://doi.org/10.1002/fld.953
  15. S. Garcia, The lid-driven square cavity flow: from stationary to time periodic and chaotic, Commun. Comput. Phys. 2 (2007), no. 5, 900-932.
  16. A. Gauthier, F. Saleri, and A. Veneziani, A fast preconditioner for the incompressible Navier-Stokes equations, Comput. Vis. Sci. 6 (2004), no. 2-3, 105-112. https://doi.org/10.1007/s00791-003-0114-z
  17. U. Ghia, K. Ghia, and C. Shin, High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), no. 3, 387-411. https://doi.org/10.1016/0021-9991(82)90058-4
  18. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.
  19. P. Gresho, D. Gartling, J. Torczynski, K. Cliffe, K. Winters, T. Garratt, A. Spence, and J. Goodrich, Is the steady viscous incompressible two-dimensional flow over a backwardfacing step at Re = 800 stable?, Internat. J. Numer. Methods Fluids 17 (1993), no. 6, 501-541. https://doi.org/10.1002/fld.1650170605
  20. P. M. Gresho and R. L. Sani, Incompressible Flow and The finite element Method, John Wiley and Sons, 1998.
  21. J. L. Guermond and L. Quartapelle, On stability and convergence of projection methods based on pressure Poisson equation, Internat. J. Numer. Methods Fluids 26 (1998), no. 9, 1039-1053. https://doi.org/10.1002/(SICI)1097-0363(19980515)26:9<1039::AID-FLD675>3.0.CO;2-U
  22. V. John, Residual a posteriori error estimates for two-level finite element methods for the Navier-Stokes equations, Appl. Numer. Math. 37 (2001), no. 4, 503-518. https://doi.org/10.1016/S0168-9274(00)00058-1
  23. D. Kay and D. Silvester, A posteriori error estimation for stabilized mixed approximations of the Stokes equations, SIAM J. Sci. Comput. 21 (1999), no. 4, 1321-1336. https://doi.org/10.1137/S1064827598333715
  24. M. Li, T. Tang, and B. Fornberg, A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 20 (1995), no. 10, 1137-1151. https://doi.org/10.1002/fld.1650201003
  25. H. Lin and S. N. Atluri, The meshless local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations, CMES Comput. Model. Eng. Sci. 2 (2001), no. 2, 117-142.
  26. S. Nicaise, L. Paquet, and Rafilipojaona, A refined mixed finite element method for stationary Navier-Stokes equations with mixed boundary conditions using Lagrange multipliers, Comput. Methods Appl. Math. 7 (2007), no. 1, 83-100.
  27. J. Oden, W. Wu, and M. Ainsworth, An a posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 111 (1994), no. 1-2, 185-202. https://doi.org/10.1016/0045-7825(94)90045-0
  28. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292-315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977. https://doi.org/10.1007/BFb0064470
  29. M. ur Rehman, C. Vuik, and G. Segal, A comparison of preconditioners for incompressible Navier-Stokes solvers, Internat. J. Numer. Methods Fluids 57 (2008), no. 12, 1731-1751. https://doi.org/10.1002/fld.1684
  30. J. Roberts and J. M. Thomas, Mixed and Hybrid Methods, Handbook of numerical analysis II, Finite element methods 1, P. Ciarlet and J. Lions, Amsterdam, 1989.
  31. R. Verfurth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309-325. https://doi.org/10.1007/BF01390056
  32. R. Verfurth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, 1996.
  33. B. I. Wohlmith and R. H. W. Hoppe, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements, Math. Comp. 68 (1999), no. 228, 1347-1378. https://doi.org/10.1090/S0025-5718-99-01125-4
  34. D. H. Wu and I. G. Currie, Analysis of a posteriori error indicator in viscous flows, Int. J. Num. Meth. Heat Fluid Flow 12 (2002), no. 3, 306-327. https://doi.org/10.1108/09615530210422983

Cited by

  1. A posteriori error estimation for incompressible viscous fluid with a new boundary condition vol.36, pp.3, 2018, https://doi.org/10.5269/bspm.v36i3.31770