×

On error estimates of some higher order penalty-projection methods for Navier-Stokes equations. (English) Zbl 0739.76017

The author first studies existing “higher order” projection schemes in the semi-discretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
65N15 Error bounds for boundary value problems involving PDEs

References:

[1] Bell, J., Colella, P., Glaz, H. (1989): A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys.85, 257-283 · Zbl 0681.76030 · doi:10.1016/0021-9991(89)90151-4
[2] Chorin, A.J. (1968): Numerical solution of the Navier-Stokes equations. Math. Comput.22, 745-762 · Zbl 0198.50103 · doi:10.1090/S0025-5718-1968-0242392-2
[3] Gresho, P.M. (1990): On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1. Theory Int. J. Numer. Methods Fluids.11, 587-620. Part 2. Implementation (with Chan, S.). Int. J. Numer. Methods Fluids11, 621-659 · Zbl 0712.76035 · doi:10.1002/fld.1650110509
[4] Heywood, J.G., Rannacher, R. (1990): 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, 275-311 · Zbl 0487.76035 · doi:10.1137/0719018
[5] Heywood, J.G., Rannacher, R. (1990): Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization. Siam. J. Numer. Anal.27, 353-384 · doi:10.1137/0727022
[6] Kim, J., Moin, P. (1985): Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys.59, 308-323 · Zbl 0582.76038 · doi:10.1016/0021-9991(85)90148-2
[7] Orszag, S.A., Israeli, M., Deville, M. (1986): Boundary conditions for incompressible flows. J. Sci. Comput.1 · Zbl 0648.76023
[8] Shen, J. (1991): Hopf bifurcation of the unsteady regularized driven cavity flows. J. Comput. Phys.95, no. 1, 228-245 · Zbl 0725.76059 · doi:10.1016/0021-9991(91)90261-I
[9] Shen, J. (1990): Long time stability and convergence for fully discrete nonlinear Galerkin methods. Appl. Anal.38, 201-229 · Zbl 0684.65095 · doi:10.1080/00036819008839963
[10] Shen, J. (1992): On error estimates of projection methods for Navier-Stokes equations: first order schemes. SIAM J. Numer. Anal.29, No. 1 (to appear) · Zbl 0741.76051
[11] Temam, R. (1968): Une m?thode d’approximation de la solution des ?quations de Navier-Stokes. Bull. Soc. Math. France96, 115-152 · Zbl 0181.18903
[12] Temam, R. (1969): Sur l’approximation de la solution des equations de Navier-Stokes par la m?thode des pas fractionnaires. II. Arch. Rat. Mech. Anal.33, 377-385 · Zbl 0207.16904 · doi:10.1007/BF00247696
[13] Temam, R. (1984): Navier-stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam · Zbl 0568.35002
[14] Temam, R. (1982): Behavior at timet=0 of the solutions of semi-linear evolution equations. J. Diff. Eqn.43, 73-92 · doi:10.1016/0022-0396(82)90075-4
[15] Temam, R. (1983): Navier-Stokes Equations and Nonlinear Functional Analysis. SIAM. CBMS · Zbl 0522.35002
[16] Temam, R. (1992): Remark on the pressure boundary condition for the projection method. Theor. Comput. Fluid Dynamics (to appear) · Zbl 0738.76054
[17] Van Kan, J. (1986): A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput.7, 870-891 · Zbl 0594.76023 · doi:10.1137/0907059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.