×

An Oseen two-level stabilized mixed finite-element method for the 2D/3D stationary Navier-Stokes equations. (English) Zbl 1237.76075

Summary: We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e., \(Q_1 - P_0\) and \(P_1 - P_0\)). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh size \(H\), a large general Stokes equation on the fine mesh with mesh size \(h = O(H)^2\). The Oseen two-level stabilized finite-element method provides an approximate solution \((u^h, p^h)\) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh size \(h\). Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15 of Springer Series in Computational Mathematics, Springer, New York, NY, USA, 1991. · Zbl 0788.73002
[2] Y. He, A. Wang, and L. Mei, “Stabilized finite-element method for the stationary Navier-Stokes equations,” Journal of Engineering Mathematics, vol. 51, no. 4, pp. 367-380, 2005. · Zbl 1069.76031
[3] Y. He and W. Sun, “Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations,” Mathematics of Computation, vol. 76, no. 257, pp. 115-136, 2007. · Zbl 1129.35004
[4] J. Li and Y. He, “A stabilized finite element method based on two local Gauss integrations for the Stokes equations,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 58-65, 2008. · Zbl 1132.35436
[5] M. A. Behr, L. P. Franca, and T. E. Tezduyar, “Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows,” Computer Methods in Applied Mechanics and Engineering, vol. 104, no. 1, pp. 31-48, 1993. · Zbl 0771.76033
[6] J. Blasco and R. Codina, “Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier-Stokes equations,” Applied Numerical Mathematics, vol. 38, no. 4, pp. 475-497, 2001. · Zbl 1011.76041
[7] R. Codina and J. Blasco, “Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations,” Numerische Mathematik, vol. 87, no. 1, pp. 59-81, 2000. · Zbl 0988.76049
[8] J. Douglas, Jr. and J. P. Wang, “An absolutely stabilized finite element method for the Stokes problem,” Mathematics of Computation, vol. 52, no. 186, pp. 495-508, 1989. · Zbl 0669.76051
[9] T. J. R. Hughes, L. P. Franca, and M. Balestra, “A new finite element formulation for computational fluid dynamics. V. Circumventing the Babu\vska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations,” Computer Methods in Applied Mechanics and Engineering, vol. 59, no. 1, pp. 85-99, 1986. · Zbl 0622.76077
[10] T. J. R. Hughes and L. P. Franca, “A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces,” Computer Methods in Applied Mechanics and Engineering, vol. 65, no. 1, pp. 85-96, 1987. · Zbl 0635.76067
[11] D. Kay and D. Silvester, “A posteriori error estimation for stabilized mixed approximations of the Stokes equations,” SIAM Journal on Scientific Computing, vol. 21, no. 4, pp. 1321-1336, 1999/00. · Zbl 0956.65100
[12] N. Kechkar and D. Silvester, “Analysis of locally stabilized mixed finite element methods for the Stokes problem,” Mathematics of Computation, vol. 58, no. 197, pp. 1-10, 1992. · Zbl 0738.76040
[13] D. J. Silvester and N. Kechkar, “Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem,” Computer Methods in Applied Mechanics and Engineering, vol. 79, no. 1, pp. 71-86, 1990. · Zbl 0706.76075
[14] P. B. Bochev, C. R. Dohrmann, and M. D. Gunzburger, “Stabilization of low-order mixed finite elements for the Stokes equations,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 82-101, 2006. · Zbl 1145.76015
[15] A. W. Wang, J. Li, and D. X. Xie, “Stabilization of the lowest-order mixed finite elements based on the local pressure projection for steady Navier-Stokes equations,” Chinese Journal of Engineering Mathematics, vol. 27, no. 2, pp. 249-257, 2010. · Zbl 1224.65230
[16] J. Xu, “A novel two-grid method for semilinear elliptic equations,” SIAM Journal on Scientific Computing, vol. 15, no. 1, pp. 231-237, 1994. · Zbl 0795.65077
[17] J. Xu, “Two-grid discretization techniques for linear and nonlinear PDEs,” SIAM Journal on Numerical Analysis, vol. 33, no. 5, pp. 1759-1777, 1996. · Zbl 0860.65119
[18] W. Layton and L. Tobiska, “A two-level method with backtracking for the Navier-Stokes equations,” SIAM Journal on Numerical Analysis, vol. 35, no. 5, pp. 2035-2054, 1998. · Zbl 0913.76050
[19] W. Layton, “A two-level discretization method for the Navier-Stokes equations,” Computers & Mathematics with Applications, vol. 26, no. 2, pp. 33-38, 1993. · Zbl 0773.76042
[20] W. Layton and W. Lenferink, “Two-level Picard and modified Picard methods for the Navier-Stokes equations,” Applied Mathematics and Computation, vol. 69, no. 2-3, pp. 263-274, 1995. · Zbl 0828.76017
[21] Y. He and A. Wang, “A simplified two-level method for the steady Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 17-18, pp. 1568-1576, 2008. · Zbl 1194.76120
[22] Y. He and K. Li, “Two-level stabilized finite element methods for the steady Navier-Stokes problem,” Computing, vol. 74, no. 4, pp. 337-351, 2005. · Zbl 1099.65111
[23] Y. He, “Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations,” SIAM Journal on Numerical Analysis, vol. 41, no. 4, pp. 1263-1285, 2003. · Zbl 1130.76365
[24] Y. N. He, “Two-level Methods based on three corrections for the 2D/3D steady Navier-Stokes equations,” International Journal of Numerical Analysis and Modeling, vol. 2, no. 1, pp. 42-56, 2011. · Zbl 1276.76041
[25] J. Li, “Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1470-1481, 2006. · Zbl 1151.76528
[26] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 1986. · Zbl 0585.65077
[27] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1977. · Zbl 0383.35057
[28] R. L. Sani, P. M. Gresho, R. L. Lee, and D. F. Griffiths, “The cause and cure of the spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations. I,” International Journal for Numerical Methods in Fluids, vol. 1, no. 1, pp. 17-43, 1981. · Zbl 0461.76021
[29] J. Li, Z. X. Chen, and Y. He, “A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier-Stokes equations,” Numerische Mathematik. In press. · Zbl 1366.76062
[30] J. Li, Y. He, and H. Xu, “A multi-level stabilized finite element method for the stationary Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 29-30, pp. 2852-2862, 2007. · Zbl 1175.76092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.