Abstract
This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 < σ = \(\tfrac{{N||f||_{ - 1} }} {{\nu ^2 }}\)≤\(\tfrac{1}{{\sqrt 2 + 1}}\), the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 < σ ≤ 5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.-Engl. Ed., 28(1), 27–35 (2007) DOI 10.1007/s10483-007-0104-x
Shang, Y. Q. and Luo, Z. D. A parallel two-level finite element method for the Navier-Stokes equations. Appl. Math. Mech.-Engl. Ed., 31(11), 1429–1438 (2010) DOI 10.1007/s10483-010-1373-7
Huang, P. Z., He, Y. N., and Feng, X. L. Two-level stabilized finite element method for Stokes eigenvalue problem. Appl. Math. Mech.-Engl. Ed., 33(5), 621–630 (2012) DOI 10.1007/s10483012-1575-7
Chen, G., Feng, M. F., and He, Y. N. Unified analysis for stabilized methods of low-order mixed finite elements for stationary Navier-Stokes equations. Appl. Math. Mech.-Engl. Ed., 34(8), 953–970 (2013) DOI 10.1007/s10483-013-1720-9
Girault, V. and Lions, J. L. Two-grid finite element scheme for the steady Navier-Stokes equations in polyhedra. Portugal. Math., 58(1), 25–57 (2001)
Girault, V. and Raviart, P. A. Finite Element Methods for the Navier-Stokes Equations, Spinger- Verlag, Berlin, 278–288 (1986)
He, Y. N. Euler implicit/explicit iterative scheme for the stationary Navier-Stokes equations. Numer. Math., 123(1), 67–96 (2013)
He, Y. N. and Li, K. T. Two-level stabilized finite element methods for the steady Navier-Stokes equations. Computing, 74(4), 337–351 (2005)
He, Y. N. and Li, J. Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 198(15), 1351–1359 (2009)
He, Y. N. and Li, J. Numerical comparisons of time-space iterative method and spatial iterative methods for the stationary Navier-Stokes equations. J. Comput. Phys., 231(20), 6790–6800 (2012)
He, Y. N., Wang, A. W., Chen, Z. X., and Li, K. T. An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier-Stokes equations. Numer. Methods PDEs, 19(6), 762–775 (2003)
Layton, W. J. A two-level discretization method for the Navier-Stokes equations. Comput. Math. Appl., 26(2), 33–38 (1993)
Layton, W. J., Lee, H. K., and Peterson, J. Numerical solution of the stationary Navier-Stokes equations using a multilevel finite element method. SIAM J. Sci. Comput., 20(1), 1–12 (1998)
Temam, R. Navier-Stokes Equations, North-Holland, Amsterdam, 158–168 (1984)
Xu, H. and He, Y. N. Some iterative finite element methods for steady Navier-Stokes equations with different viscosities. J. Comput. Phys., 232(1), 136–152 (2013)
Girault, V. and Lions, J. L. Two-grid finite element scheme for the transient Navier-Stokes problem. Math. Model. Numer. Anal., 35(5), 945–980 (2001)
He, Y. N. Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal., 41(4), 1263–1285 (2004)
He, Y. N. and Li, K. T. Asymptotic behavior and time discretization analysis for the nonstationary Navier-Stokes problem. Numer. Math., 98(4), 647–673 (2004)
He, Y. N. and Sun, W. W. Stability and convergence of the Crank-Nicolson/ Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal., 45(2), 837–869 (2007)
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), 275–311 (1982)
Heywood, J. G. and Rannacher, R. Finite element approximations of the nonstationary Navier- Stokes problem, II: stability of the solution and error estimates uniform in time. SIAM J. Numer. Anal., 23(4), 750–777 (1986)
Heywood, J. G. and Rannacher, R. Finite element approximations of the nonstationary Navier- Stokes problem, III: smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal., 25(3), 489–512 (1988)
Hill, A. T. and Süli, E. Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal., 20(4), 633–667 (2000)
Labovsky, A., Layton, W. J., Manica, C. C., Neda, M., and Rebholz, L. G. The stabilized extrapolated trapezoidal finite element method for the Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 198(9), 958–974 (2009)
Li, K. T., Huang, A. X., and Huang, Q. H. Finite Element Methods and Their Applications (in Chinese), Academic Press, Beijing, 344–347 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 11271298)
Rights and permissions
About this article
Cite this article
Zhang, G., Dong, X., An, Y. et al. New conditions of stability and convergence of Stokes and Newton iterations for Navier-Stokes equations. Appl. Math. Mech.-Engl. Ed. 36, 863–872 (2015). https://doi.org/10.1007/s10483-015-1953-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-015-1953-9