Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations
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Abstract:
A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair $(X_h,M_h)$ which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters $\epsilon ,~\Delta t$ and $h$ are sufficiently small.References
- M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér. 12 (1978), no. 3, 211–236, iii (English, with French summary). MR 509973, DOI 10.1051/m2an/1978120302111
- B. Brefort, J.-M. Ghidaglia, and R. Temam, Attractors for the penalized Navier-Stokes equations, SIAM J. Math. Anal. 19 (1988), no. 1, 1–21. MR 924541, DOI 10.1137/0519001
- F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, Efficient solutions of elliptic systems (Kiel, 1984) Notes Numer. Fluid Mech., vol. 10, Friedr. Vieweg, Braunschweig, 1984, pp. 11–19. MR 804083
- Alexandre Joel Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745–762. MR 242392, DOI 10.1090/S0025-5718-1968-0242392-2
- Alexandre Joel Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969), 341–353. MR 242393, DOI 10.1090/S0025-5718-1969-0242393-5
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
- Yinnian He, A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem, IMA J. Numer. Anal. 23 (2003), no. 4, 665–691. MR 2011345, DOI 10.1093/imanum/23.4.665
- Yinnian He, Yanping Lin and Weiwei Sun, Stabilized finite element method for the Navier-Stokes problem, submitted.
- John G. Heywood and Rolf 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), no. 2, 275–311. MR 650052, DOI 10.1137/0719018
- John G. Heywood and Rolf Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. MR 1043610, DOI 10.1137/0727022
- Ai Xiang Huang and Kai Tai Li, A penalty method for nonstationary Navier-Stokes equations, Acta Math. Appl. Sinica 17 (1994), no. 3, 473–480 (Chinese). MR 1333935
- Thomas J. R. Hughes, Wing Kam Liu, and Alec Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, J. Comput. Phys. 30 (1979), no. 1, 1–60. MR 524162, DOI 10.1016/0021-9991(79)90086-X
- Nasserdine Kechkar and David Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem, Math. Comp. 58 (1992), no. 197, 1–10. MR 1106973, DOI 10.1090/S0025-5718-1992-1106973-X
- W. Layton and L. Tobiska, A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), no. 5, 2035–2054. MR 1639994, DOI 10.1137/S003614299630230X
- Jie Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal. 32 (1995), no. 2, 386–403. MR 1324294, DOI 10.1137/0732016
- Jie Shen, On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations, Numer. Math. 62 (1992), no. 1, 49–73. MR 1159045, DOI 10.1007/BF01396220
- Jie Shen, On error estimates of projection methods for Navier-Stokes equations: first-order schemes, SIAM J. Numer. Anal. 29 (1992), no. 1, 57–77. MR 1149084, DOI 10.1137/0729004
- Roger Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France 96 (1968), 115–152 (French). MR 237972
- R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I, Arch. Rational Mech. Anal. 32 (1969), 135–153 (French). MR 237973, DOI 10.1007/BF00247678
- R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II, Arch. Rational Mech. Anal. 33 (1969), 377–385 (French). MR 244654, DOI 10.1007/BF00247696
Additional Information
- Yinnian He
- Affiliation: Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
- Email: heyn@mail.xjtu.edu.cn
- Received by editor(s): July 2, 2003
- Received by editor(s) in revised form: May 15, 2004
- Published electronically: February 16, 2005
- Additional Notes: This work was subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 1201-1216
- MSC (2000): Primary 35L70, 65N30, 76D06
- DOI: https://doi.org/10.1090/S0025-5718-05-01751-5
- MathSciNet review: 2136999