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OPTIMAL ERROR ESTIMATE FOR SEMI-DISCRETE GAUGE-UZAWA METHOD FOR THE NAVIER-STOKES EQUATIONS

  • Pyo, Jae-Hong (DEPARTMENT OF MATHEMATICS KANGWON NATIONAL UNIVERSITY)
  • Published : 2009.07.31

Abstract

The gauge-Uzawa method which has been constructed in [11] is a projection type method to solve the evolution Navier-Stokes equations. The method overcomes many shortcomings of projection methods and displays superior numerical performance [11, 12, 15, 16]. However, we have obtained only suboptimal accuracy via the energy estimate in [11]. In this paper, we study semi-discrete gauge-Uzawa method to prove optimal accuracy via energy estimate. The main key in this proof is to construct the intermediate equation which is formed to gauge-Uzawa algorithm. We will estimate velocity errors via comparing with the intermediate equation and then evaluate pressure errors via subtracting gauge-Uzawa algorithm from Navier-Stokes equations.

Keywords

References

  1. D. L. Brown, R. Cortez, and M. L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys. 168 (2001), no. 2, 464–499 https://doi.org/10.1006/jcph.2001.6715
  2. A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22(1968), 745–762
  3. P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, 1988
  4. M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no.1, 74–97 https://doi.org/10.1137/0520006
  5. Weinan E. and J.-G. Liu, Gauge method for viscous incompressible flows, Comm. Math. Sci. 1 (2003), 317–332
  6. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986
  7. J. G. Heywood and R. Rannacher, Finite 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 (1982), no. 2, 275–311
  8. R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Analysis 21 (1976), no. 4, 397–431 https://doi.org/10.1016/0022-1236(76)90035-5
  9. R. H. Nochetto and J.-H. Pyo, Optimal relaxation parameter for the Uzawa method, Numer. Math. 98 (2004), no. 4, 695–702 https://doi.org/10.1007/s00211-004-0522-0
  10. R. H. Nochetto and J.-H. Pyo, Error estimates for semi-discrete gauge methods for the Navier-Stokes equations, Math. Comp. 74 (2005), no. 250, 521–542 https://doi.org/10.1090/S0025-5718-04-01687-4
  11. R. H. Nochetto and J.-H. Pyo, The gauge-Uzawa finite element method. I. The Navier-Stokes equations, SIAM J. Numer. Anal. 43 (2005), no. 3, 1043–1068 https://doi.org/10.1137/040609756
  12. R. H. Nochetto and J.-H. Pyo, The gauge-Uzawa finite element method. II. The Boussinesq equations, Math. Models Methods Appl. Sci. 16 (2006), no. 10, 1599–1626
  13. A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Advances in Numerical Mathematics. B. G. Teubner, Stuttgart, 1997
  14. J.-H. Pyo, The gauge-Uzawa and related projection finite element methods for the evolution Navier-Stokes equations, Ph.D. dissertation, University of Maryland, 2002
  15. J.-H. Pyo and J. Shen, Normal mode analysis of second-order projection methods for incompressible flows, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 3, 817–840
  16. J.-H. Pyo and J. Shen, Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Phys. 221 (2007), no. 1, 181–197 https://doi.org/10.1016/j.jcp.2006.06.013
  17. R. T´emam, Sur l'approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires. II, Arch. Rational Mech. Anal. 33 (1969), 377–385
  18. C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow, Math. Comp. 69 (2000), no. 232, 1385–1407 https://doi.org/10.1090/S0025-5718-00-01248-5

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