Multi-level spectral Galerkin method for the Navier-Stokes equations, II: Time discretization

Advances in Computational Mathematics

Volumn 25, Issue 4, 2006, Pages 403-433

  He, Yinnian ^a   Liu, Kam Moon ^b  

^a XI'AN JIAOTONG UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Error estimate; Multi level; Navier Stokes problem; Spectral Galerkin method

Indexed keywords

EID: 33748709839     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-004-7640-1     Document Type: Article

References (26)

1. C. Devulder, M. Marion and E.S. Titi, On the rate of convergence of the nonlinear Galerkin methods, Math. Comp. 60 (1993) 495-514.
2. C. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, KAIRO Model. Math. Anal. Numer. 22 (1988) 93-114.
3. B. Garcia-Archilla, J. Novo and E.S. Titi, Postprocessing the Galerkin method: A novel approach to approximate inertial manifolds, SIAM J. Numer. Anal. 35 (1998) 941-972.
4. B. Garcia-Archilla, J. Novo and E.S. Titi, An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations, Math. Comp. 68 (1999) 893-911.
5. V. Girault and J.L. Lions, Two-grid finite element scheme for the steady Navier-Stokes equations in polyhedra, Portugal. Math. 58 (2001) 25-57.
6. V. Girault and J.L. Lions, Two-grid finite element scheme for the transient Navier-Stokes problem, Math. Model. Numer. Anal. 35 (2001) 945-980.
7. V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations (Springer, Berlin/New York, 1979).
8. Y. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal. 41 (4) (2003) 1263-1285.
9. Y. He and K. Li, Stability and convergence of finite element nonlinear Galerkin methods for the Navier-Stokes equations, Numer. Math. 79 (1998) 77-106.
10. Y. He and K. Li, Postprocessing techniques for the Navier-Stokes equations, in: Dynamics of Continuous, Discrete and Impulsive Systems, to appear.
11. Y. He, K.M. Liu and W. Sun, Multi-level spectral Galerkin method for the Navier-Stokes problem I: Spatial discretization, submitted.
12. Y. He and R.M.M. Mattheij, Stability and convergence for the reform postprocessing Galerkin method, Nonlinear Anal. Real. World Appl. 1 (2000) 517-533.
13. Y. He and R.M.M. Mattheij, Reform postprocessing Galerkin method for the Navier-Stokes equations, submitted.
14. J.G. Heywood and R. Rannacher, On the question of turbulence modeling by appoximate inertial manifolds and the nonlinear Galerkin method, SIAM J. Numer. Anal. 30 (1993) 1603-1621.
15. 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) 275-311.
16. W. Layton, A two-level discretization method for the Navier-Stokes equations, Comput. Math. Appl. 26 (1993) 33-38.
17. W. Layton, H.K. Lee and J. Peterson, Numerical solution of the stationary Navier-Stokes equations using a multilevel finite element method, SIAM J. Sci. Comput. 20 (1998) 1-12.
18. W. Layton and H.W.J. Leferink, A multilevel mesh independence principle for the Navier-Stokes equations, SIAM J. Numer. Anal. 33 (1996) 17-30.
19. W. Layton and L. Tobiska, A two-level method with backtraking for the Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998) 2035-2054.
20. L.G. Margolin, E.S. Titi and S. Wynne, Postprocessing the Galerkin and nonlinear Galerkin methods - a truncation analysis point of view, SIAM J. Numer. Anal., to appear.
21. G. Métivier, Étude asymptotique des valeurs propres et la fonction spectrale de problems aux limites, Thèse, Université de Nice, France (1976).
22. M.A. Olshanskii, Two-level method and some a priori estimates in unsteady Navier-Stokes calculations, J. Comput. Appl. Math. 104 (1999) 173-191.
23. J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal. 38 (1990) 201-229.
24. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Springer, New York/Berlin, 1988).
25. J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994) 231-237.
26. J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996) 1759-1777.