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Rannacher, Rolf

Numerical analysis of the Navier-Stokes equations. (English) Zbl 0798.76041

Appl. Math., Praha 38, No. 4-5, 361-380 (1993).
The author surveys the state of the art in the field of Galerkin methods for the incompressible Navier-Stokes equations. The key issues are, as stated by the author: 1. The treatment of the incompressibility constraint by Chorin’s projection method: The problem of the nonphysical pressure boundary conditions. 2. The choice of outflow boundary conditions: The problem of well-posedness. 3. The modeling of the nonlinearity: The mechanism behind the success of the nonlinear Galerkin method.
Reviewer: G.Warnecke (Stuttgart)

Cited in 8 Documents

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Keywords:

incompressibility constraint; Chorin’s projection method; boundary conditions; well-posedness; nonlinear Galerkin method
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\textit{R. Rannacher}, Appl. Math., Praha 38, No. 4--5, 361--380 (1993; Zbl 0798.76041)
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References:

[1] Blum H.: Asymptotic Error Expansion and Defect Correction in the Finite Element Method. Habilitationsschrift, Universität Heidelberg, 1991.
[2] Brezzi F., and J. Pitkäranta: On the stabilization of finite element approximations of the Stokes equations. Efficient Solution of Elliptic Systems (W. Hackbush, Vieweg, Braunschweig, 1984.
[3] Bristeau M. O., Glowinski R., and J. Periaux: Numerical methods for the Navier-Stokes equations: Applications to the simulation of compressible and incompressible viscous flows. In Computer Physics Report, Research Report UH/MD-4, University of Houston, 1987. · Zbl 0675.76030
[4] Chorin A. J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22 (1968), 745-762. · Zbl 0198.50103
[5] Devulder C., Marion M., and E. S.Titi: On the rate of convergence of the nonlinear Galerkin methods. to appear in Math. Comp.. · Zbl 0783.65053
[6] Foias C., Manley O., and R. Temam: Modelling of the interaction of small and large eddies in two dimensional turbulent flows. \(M^2\) AN 22 (1988), 93-114. · Zbl 0663.76054
[7] Girault V., and P. A. Raviart: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin-Heidelberg, 1986.
[8] Gresho P. M., and S. T. Chan: 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, Part 2: Implementation. Int. J. Numer. Meth. in Fluids 11 (1990), 587-620, 621-659. · Zbl 0712.76035
[9] Gresho P. M.: Some current CFD issues relevant to the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng. 87(1991), 201-252. · Zbl 0760.76018
[10] Harig J.: A 3-d finite element upwind approximation of the stationary Navier Stokes equations. IWR-Report, Universität Heidelberg, 1991. · Zbl 0782.76056
[11] Heywood J. G.: On uniqueness questions in the theory of viscous flow. Acta math. 136 (1976), 61-102. · Zbl 0347.76016
[12] Heywood J .G., R. Rannacher: Finite element approximation of the nonstationary Navier-Stokes problem. Part 1: regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal 19 (1982), 275-311 · Zbl 0487.76035
[13] Heywood J. G., R. Rannacher: Finite element approximation of the nonstationary Navier-Stokes problem. Part 2: Stability of solutions and error estimates uniform in time. ibidem 23, (1986), 750-777 · Zbl 0611.76036
[14] Heywood J.G., R. Rannacher: Finite element approximation of the nonstationary Navier-Stokes problem. Part 3: Smoothing property and higher order error estimates for spatial discretization. ibidem 25 (1988), 489-512 · Zbl 0646.76036
[15] Heywood J.G., R. Rannacher: Finite element approximation of the nonstationary Navier-Stokes problem. Part 4: error analysis for second-order time discretization. ibidem 27 (1990), 353-384. · Zbl 0694.76014
[16] Heywood J. G., Rannacher R., and S.Turek: Artifical boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Universität Heidelberg, April 1992, Preprint.
[17] Heywood J. G., R. Rannacher: On the question of turbulence modeling by approximate inertial manifolds and the nonlinear Galerkin method. Universität Heidelberg, May 1992, Preprint. · Zbl 0791.76042
[18] Hughes T. J. R., Franca L. P., and M. Balestra: A new finite element formulation for computational fluid mechanics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommoding equal order interpolation. Comp. Meth. Appl. Mech. Eng. 59 (1986), 85-99. · Zbl 0622.76077
[19] Kracmar S., and J. Neustupa: Global existence of weak solutions of a nonstationary variational inequality of the Navier-Stokes type with mixed boundary conditions. Dept. of Techn. Math., Czech Techn. Univ, Praha, 1992, Preprint.
[20] Marion M., and R. Temam: Nonlinear Galerkin methods: The finite element case. Numer. Math. 57, (1990), 205-226. · Zbl 0702.65081
[21] Prohl A., and R. Rannacher: On some pseudo-compressibility methods for the Navier-Stokes equations. in preparation.
[22] Rannacher R.: Numerical analysis of nonstationary fluid flow (a survey). Applications of Mathematics in Industry and Technology (V. C. Boffi and H. Neunzert, Teubner, Stuttgart, 1989, pp. 34-53.
[23] Rannacher R.: On Chorin’s projection method for the incompressible Navier-Stokes equations. Springer, (R. Rautmann, et.al., Proc. Oberwolfach Conf., 19.-23. 8. 1991.
[24] Rannacher R., and S. Turek: Simple nonconforming quadrilateral Stokes elements. Numer. Meth. Part. Diff. Equ. 8 (1992), 97-111. · Zbl 0742.76051
[25] Shen J.: On error estimates of projection methods for the Navier-Stokes equations: First order schemes. SIAM J. Numer. Anal. (1991).
[26] Shen J.: On error estimates of higher order projection and penalty-projection methods for Navier-Stokes equations. Dept. of Math., Indiana University, 1992, Preprint. · Zbl 0782.76025
[27] Turek S.: Tools for simulating nonstationary incompressible flow via discretely divergence-free finite element models. Universität Heidelberg, May 1992, Preprint. · Zbl 0794.76051
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