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A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations

Published online by Cambridge University Press:  09 September 2015

Yueqiang Shang  and
Jin Qin
Yueqiang Shang*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Jin Qin
Affiliation:
School of Mathematics and Computer Science, Zunyi Normal College, Zunyi 563002, China
*
*Corresponding author. Email: [email protected] (Y. Q. Shang), [email protected] (J. Qin)
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Abstract

Based on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.

Keywords

65N5576D0776M10Navier-Stokes equationsfinite elementtwo-level methodparallel algorithm
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 6 , December 2015 , pp. 715 - 735
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Xu, J. C. and Zhou, A. H., Local and parallel finite element algorithms based on two-grid discretizations, Math. Comput., 69 (2000), pp. 881–909.Google Scholar
[2]Xu, J. C. and Zhou, A. H., Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math., 14 (2001), pp. 293–327.Google Scholar
[3]He, Y. N., Xu, J. C. and Zhou, A. H., Local and parallel finite element algorithms for the Navier-Stokes problem, J. Comput. Math., 24(3) (2006), pp. 227–238.Google Scholar
[4]Ma, Y. C., Zhang, Z. M. and Ren, C. F., Local and parallel finite element algorithms based on two-grid discretization for the stream function form of Navier-Stokes equations, Appl. Math. Comput., 175 (2006), pp. 786–813.Google Scholar
[5]Ma, F. Y., Ma, Y. C. and Wo, W. F., Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations, Appl. Math. Mech., 28(1) (2007), pp. 27–35.Google Scholar
[6]Shang, Y. Q., A parallel two-level linearization method for incompressible flow problems, Appl. Math. Lett., 24 (2011), pp. 364–369.Google Scholar
[7]Shang, Y. Q., He, Y. N., Kim, D. W. and Zhou, X. J., A new parallel finite element algorithm for the stationary Navier-Stokes equations, Finite Elem. Anal. Des., 47 (2011), pp. 1262–1279.Google Scholar
[8]Shang, Y. Q., He, Y. N. and Luo, Z. D., A comparison of three kinds of local and parallel finite element algorithms based on two-grid discretizations for the stationary Navier-Stokes equations, Comput. Fluids, 40 (2011), pp. 249–257.Google Scholar
[9]He, Y. N., Mei, L. Q., Shang, Y. Q. and Cui, J., Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations, J. Sci. Comput., 44(1) (2010), pp. 92–106.Google Scholar
[10]Shang, Y. Q. and He, Y. N., A parallel Oseen-linearized algorithm for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 209-212 (2012), pp. 172–183.Google Scholar
[11]Shang, Y. Q., A parallel two-level finite element variational multiscale method for the Navier-Stokes equations, Nonlinear Anal., 84 (2013), pp. 103–116.Google Scholar
[12]Shang, Y. Q. and Huang, S. M., A parallel subgrid stabilized finite element method based on two-grid discretization for simulation of 2D/3D steady incompressible flows, J. Sci. Comput., 60 (2014), pp. 564–583.Google Scholar
[13]Adams, R., Sobolev Spaces, Academic Press Inc., New York, 1975.Google Scholar
[14]Heywood, J. G. and Rannacher, R., 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(2) (1982), pp. 275–311.Google Scholar
[15]Girault, V. and Raviart, P. A., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986.CrossRefGoogle Scholar
[16]Temam, R., Navier-Stokes Equations, North-Holland, Amsterdam, 1984.Google Scholar
[17]Shang, Y. Q. and He, Y. N., Parallel iterative finite element algorithms based on full domain partition for the stationary Navier-Stokes equations, Appl. Numer. Math., 60(7) (2010), pp. 719–737.Google Scholar
[18]Shang, Y. Q. and Wang, K., Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations, Numer. Algor., 54 (2010), pp. 195–218.Google Scholar
[19]Arnold, D. N. and Liu, X., Local error estimates for finite element discretizations of the Stokes equations, RAIRO M2AN, 29 (1995), pp. 367–389.CrossRefGoogle Scholar
[20]Arnold, D. N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations, Calc., 21 (1984), pp. 337–344.Google Scholar
[21]Fortin, M., Calcul Numérique des Ecoulements Fluides de Bingham et des Fluides Newtoniens Incompressible par des Méthodes D’eléments Finis, Doctoral thesis, Université de Paris VI, 1972.Google Scholar
[22]Hood, P. and Taylor, C., A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids, 1 (1973), pp. 73–100.Google Scholar
[23]Crouzeix, M. and Raviart, P. A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numér., 7(R-3) (1973), pp. 33–76.Google Scholar
[24]Mansfield, L., Finite element subspaces with optimal rates of convergence for stationary Stokes problem, RAIRO Anal. Numér., 16 (1982), pp. 49–66.CrossRefGoogle Scholar
[25]Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flows, Gordon and Breach, New York, 1969.Google Scholar
[26]He, Y. N. and Li, J., Convergence of three iterative methods based on finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 198 (2009), pp. 1351–1359.Google Scholar
[27]Xu, H. and He, Y. N., Some iterative finite element methods for steady Navier-Stokes equations with different viscosities, J. Comput. Phys., 232 (2013), pp. 136–152.Google Scholar