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Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations. (English) Zbl 1065.35025

Penalty numerical methods for time-dependent Navier-Stokes equations in a two-dimensional region are investigated. The penalty method is based on the idea of replacing the original Navier-Stokes equation by a modification of the form \[ u_{\varepsilon t} -\nu \Delta u_\varepsilon+ (u \cdot \nabla) v +{1\over2} (\text{div}\, u)v +\nabla p_\varepsilon =0 \] and the incompressibility constraint \( \text{div}\, u =0\) by an equation of the form \[ \text{div}\, u_\varepsilon + {\varepsilon\over\nu} p_\varepsilon =0, \] where \( u_\varepsilon \) and \( p_\varepsilon \) represent penalty velocity and pressure, respectively, which are the approximation of real velocity and pressure \(u,p\). The well known results for this method consist of convergence and error estimates of a couple \( (u_\varepsilon, p_\varepsilon)\) to the solution of Navier-Stokes equations \((u,p)\) for \(\varepsilon \rightarrow 0\) and error estimates for backward Euler semi-discretization in time were also proved. Those works are extended in this paper to the case of the fully discrete penalty finite element method. First, some regularity results for the time discretized penalty Navier-Stokes equation with the Euler backward scheme is proved. Then for the fully discrete finite element method of the penalty Navier-Stokes equations some boundedness results for the numerical solution \(( u^n_{\varepsilon h}, p^n_{\varepsilon h}) \) are presented. These results are used for proving the main aim of this paper: to derive optimal error estimates for the fully discrete penalty finite element method.

MSC:

35A35 Theoretical approximation in context of PDEs
35Q30 Navier-Stokes equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D06 Statistical solutions of Navier-Stokes and related equations
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[1] 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). · Zbl 0428.65059
[2] 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. · Zbl 0696.35131 · doi:10.1137/0519001
[3] 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.
[4] Alexandre Joel Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745 – 762. · Zbl 0198.50103
[5] Alexandre Joel Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969), 341 – 353. · Zbl 0184.20103
[6] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[7] 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. · Zbl 0585.65077
[8] 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. · Zbl 1135.76331 · doi:10.1093/imanum/23.4.665
[9] YINNIAN HE, YANPING LIN AND WEIWEI SUN, Stabilized finite element method for the Navier-Stokes problem, submitted. · Zbl 1089.76033
[10] 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. · Zbl 0487.76035 · doi:10.1137/0719018
[11] 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. · Zbl 0694.76014 · doi:10.1137/0727022
[12] 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).
[13] 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. · Zbl 0412.76023 · doi:10.1016/0021-9991(79)90086-X
[14] 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. · Zbl 0738.76040
[15] 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. · Zbl 0913.76050 · doi:10.1137/S003614299630230X
[16] Jie Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal. 32 (1995), no. 2, 386 – 403. · Zbl 0822.35008 · doi:10.1137/0732016
[17] 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. · Zbl 0782.76025 · doi:10.1007/BF01396220
[18] 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. · Zbl 0741.76051 · doi:10.1137/0729004
[19] Roger Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France 96 (1968), 115 – 152 (French). · Zbl 0181.18903
[20] 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). · Zbl 0195.46001 · doi:10.1007/BF00247678
[21] 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). · Zbl 0207.16904 · doi:10.1007/BF00247696
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