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Section B

Two-level stabilized finite element method for the transient Navier–Stokes equations

, &
Pages 2341-2360 | Received 23 Jun 2008, Accepted 11 Nov 2008, Published online: 20 May 2010

References

  • Adams , R. 1975 . Sobolev Spaces , New York : Academic Press .
  • Ammi , A. and Marion , M. 1994 . Nonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier–Stokes equations . Numer. Math. , 68 : 189 – 213 .
  • Bramble , J. and Pasciak , J. 1988 . A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems . Math. Comp. , 50 : 1 – 17 .
  • Bramble , J. and Xu , J. 1991 . Some estimates for a weighted L2 projection . Math. Comp. , 56 : 463 – 476 .
  • Brefort , B. , Ghidaglia , J. and Temam , R. 1988 . Attractor for the penalty Navier–Stokes equations . SIAM J. Numer. Anal. , 19 : 1 – 21 .
  • Ciarlet , P. 1978 . “ The Finite Element Method for Elliptic Problems ” . Amsterdam : North-Holland .
  • Douglas , J. and Wang , J. 1989 . An absolutely stabilized finite element method for the Stokes problem . Math. Comp. , 52 : 495 – 508 .
  • Ervin , V. , Layton , W. and Maubach , J. 1996 . A posteriori error estimators for a two-level finite element method for the Navier–Stokes equations . Numer. Methods Partial Differential Equations , 12 : 333 – 346 .
  • Giraut , V. and Raviart , P. 1979 . “ Finite Element Approximation of the Navier–Stokes Equations ” . Berlin : Springer-Verlag .
  • He , Y. 2003 . A fully discrete stabilized finite-element method for the time-dependent Navier–Stokes problem . IMA J. Numer. Anal. , 23 : 665 – 691 .
  • He , Y. 2004 . A two-level finite element Galerkin method for the nonstationary Navier–Stokes equations I: Spatial discretization . J. Comput. Math. , 22 : 21 – 32 .
  • He , Y. and Li , K. 2005 . Two-level stabilized finite element methods for the steady Navier–Stokes problem . Computing , 74 : 337 – 351 .
  • He , Y. , Lin , Y. and Sun , W. 2006 . Stabilized finite element method for the non-stationary Navier–Stokes problem . Discrete Contin. Dyn. Syst. Ser. B , 6 : 41 – 68 .
  • He , Y. and Sun , W. 2007 . Stabilized finite element method based on the Crank–Nicolson extrapolation scheme for the time-dependent Navier–Stokes equations . Math. Comp. , 76 : 115 – 136 .
  • He , Y. , Wang , A. and Mei , L. 2005 . Stabilized finite-element method for the stationary Navier–Stokes equations . J. Engrg. Math. , 51 : 367 – 380 .
  • Heywood , J. and Rannacher , R. 1988 . Finite element approximation of the nonstationary Navier–Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization . SIAM J. Numer. Anal. , 25 : 489 – 512 .
  • Heywood , J. and Rannacher , R. 1990 . Finite-element approximation of the nonstationary Navier–Stokes problem part IV: Error analysis for second-order time discretization . SIAM J. Numer. Anal. , 27 : 353 – 384 .
  • Hill , A. and Süli , E. 2000 . Approximation of the global attractor for the incompressible Navier–Stokes equations . IMA J. Numer. Anal. , 20 : 633 – 667 .
  • Hughes , T. and Franca , L. 1987 . A new finite element formulation for CFD. VII: The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity, pressure spaces . Comput. Methods Appl. Mech. Engrg. , 65 : 85 – 97 .
  • Kay , D. and Silvester , D. 2000 . A posteriori error estimation for stabilized mixed approximations of the Stokes equations . SIAM J. Sci. Comput. , 21 : 1321 – 1337 .
  • Kechkar , N. and Silvester , D. 1992 . Analysis of locally stabilized mixed finite element methods for the Stokes problem . Math. Comp. , 58 : 1 – 10 .
  • Layton , W. 1993 . A two level discretization method for the Navier–Stokes equations . Comput. Math. Appl. , 26 : 33 – 38 .
  • Layton , W. and Lenferink , W. 1995 . Two-grid Picard, defect correction for the Navier–Stokes equations . Appl. Math. Comput. , 80 : 1 – 12 .
  • Layton , W. and Tobiska , L. 1998 . A two-level method with backtracking for the Navier–Stokes equations . SIAM J. Numer. Anal. , 35 : 2035 – 2054 .
  • Li , J. and He , Y. 2008 . A stabilized finite element method based on two local Gauss integrations for the Stokes equations . J. Comput. Appl. Math. , 214 : 58 – 65 .
  • Li , J. , He , Y. and Chen , Z. 2007 . A new stabilized finite element method for the transient Navier–Stokes equations . Comput. Methods Appl. Mech. Engrg. , 197 : 22 – 35 .
  • Shang , Y. New stabilized finite element methods for the time-dependent Stokes problem . to appear in Internat. J. Numer. Methods Fluids ,
  • Silvester , D. 1994 . Optimal low-order finite element methods for incompressible flow . Comput. Methods Appl. Mech. Engrg. , 111 : 357 – 368 .
  • Silvester , D. and Kechkar , N. 1990 . Stabilized bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem . Comput. Methods Appl. Mech. Engrg. , 79 : 71 – 87 .
  • Temam , R. 1984 . “ Navier–Stokes Equations, Theory and Numerical Analysis ” . Amsterdam : North-Holland .
  • Verfürth , R. 1984 . A multilevel algorithm for mixed problems . SIAM J. Numer. Anal. , 21 : 264 – 271 .
  • Xu , J. 1994 . A novel two-grid method for semilinear elliptic equations . SIAM J. Sci. Comput. , 15 : 231 – 237 .
  • Xu , J. 1996 . Two-grid discretization techniques for linear and nonlinear PDEs . SIAM J. Numer. Anal. , 33 : 1759 – 1777 .

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