An analysis of a mixed finite element method for the Navier-Stokes equations. (English) Zbl 0396.65070

Girault, V.; Raviart, P. A.

Numer. Math. 33, 235-271 (1979).

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids
65Z05	Applications to the sciences

Keywords:

Nonsingular Solution; Convergence; Quasioptimal Error Bounds; Discrete Compactness; Navier-Stokes Equations; Divergence; Finite Element Method; Stream Function; Incompressibility; Vorticity

References:

[1]	Argyris, J.H., Mareczek, G.: Finite element analysis of slow incompressible viscous fluid motion. Ing.43, 92-109 (1974) · Zbl 0274.76028
[2]	Baker, A.J.: Finite element solution algorithm for viscous incompressible fluid dynamics. Internat. J. Numer. Methods Engrg.6, 89-101 (1973) · Zbl 0255.76042
[3]	Bercovier, M., Pironneau, O.: Comptes rendus.285, série A, 1085 (1977)
[4]	Brebbia, C.A., Smith, S.L.: Finite element solution of Navier-Stokes equations for transient two-dimensional incompressible flow. J. Computational Phys.17, 235-245 (1975) · Zbl 0302.76017
[5]	Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO8-R2, 129-151 (1974) · Zbl 0338.90047
[6]	Brezzi, F.: Finite element approximation of the Von Karmán equations. Numer. Math. (to appear).
[7]	Cheng, R.T.: Numerical solution of the Navier-Stokes equations by the finite element method. Phys. Fluids15, 2098-2105 (1972) · Zbl 0252.76017
[8]	Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[9]	Ciarlet, P.G., Raviart, P.A.: Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg.1, 217-249 (1972) · Zbl 0261.65079
[10]	Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for the biharmonic equation. Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor ed. New York-London: Academic Press, 125-145 1974
[11]	Crouzeix, M., Raviart, P.A.: Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIROR.3., 33-76 (1973) · Zbl 0302.65087
[12]	Fortin, M.: Calcul numérique des écoulements des fluides de Bingham et des fluides Newtoniens incompressibles par la méthode des éléments finis. Thèse Paris (1972)
[13]	Gallagher, R.H., Oden, J.T., Taylor, C., Zienkiewicz, O.C. (eds.): Finite Elements in Fluids. New York: Wiley 1975 · Zbl 0341.76001
[14]	Gartling, D.K., Becker, E.B.: Finite element analysis of viscous incompressible fluid flow I. Comput. Methods Appl. Mech. Engrg.8, 51-60 (1976) · Zbl 0325.76036
[15]	Girault, V.: A combined finite element and Marker-and-Cell method for solving Navier-Stokes equations. Numer. Math.26, 39-59 (1976) · Zbl 0313.65105
[16]	Girault, V.: A mixed finite element method for the stationary Stokes equations. SIAM J. Numer. Anal.15,o 3, 534-555 (1978) · Zbl 0387.76029
[17]	Glowinski, R.: Approximations externes, par éléments finis de Lagrange d’ordre un et deux, du problème de Dirichlet pour l’opérateur biharmonique. Méthode itérative de résolution des problèmes approchés. Topics in Numerical Analysis, J. Miller ed., 123-171. New York-London: Academic Press 1973
[18]	Glowinski, R., Pironneau, O.: Comptes rendus.282, série A, 223, 1315 (1976)
[19]	Grisvard, P.: Singularité des solutions du problème de Stokes dans un polygone. Séminaires d’Analyse Numérique (1978), Paris. (To appear)
[20]	Jamet, P., Raviart, P.A.: Numerical solution of the stationary Navier-Stokes equations by finite element methods. Lecture Notes in Computer Science10, 193-223. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0285.76007
[21]	Johnson, C.: A mixed finite element method for the Navier-Stokes equations, RAIRO (to appear)
[22]	Keller, H.B.: Approximation methods for non linear problems with applications to two-points boundary value problems. Math. Comput.29, 464-474 (1975) · Zbl 0308.65039
[23]	Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach 1969 · Zbl 0184.52603
[24]	Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod 1969
[25]	Miyoshi, T.: A mixed finite element method for the solution of the Von Karmán equations. Numer. Math.26, 255-269 (1976) · Zbl 0315.65064
[26]	Olson, M.D.: A variational finite element method for two-dimensional steady viscous flows. Proceedings of the Special Conference on Finite Element Methods in Civil Engineering, 585-616 (1972)
[27]	Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems, Lecture Notes in Mathematics606, 292-315. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0362.65089
[28]	Scholz, R.: Approximation von Sattelpunkten mit finiten Elementen. Bonn. Math. Schr.89, 53-66 (1976) · Zbl 0359.65096
[29]	Scholz, R.: A mixed method for 4th order problems using linear finite elements. RAIRO12 no 1, 85-90 (1978) · Zbl 0382.65059
[30]	Scott, R.: OptimalL ? estimates for the finite element method on irregular meshes. Math. Comput.30, no 136, 681-697 (1976) · Zbl 0349.65060
[31]	Second International Symposium on Finite Element Methods in Flow Problems. S. Margherita Ligure (1978)
[32]	Le Tallec, P.: Thèse de 3ème cycle, Paris.
[33]	Taylor, C., Hood, P.: A numerical solution of the Navier-Stokes equations using the finite element technique. Computer and Fluids1, 73-100 (1973) · Zbl 0328.76020
[34]	Teman, R.: Navier-Stokes Equations Amsterdam: North-Holland 1977
[35]	Thomas, J.M.: Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Thèse Paris (1977)
[36]	Welch, J.E.: Harlow, F.H., Shannon, J.P., Daly, B.J.: The MAC method, LASL Report no LA-3425, Los Alamos Scientific Laboratory, Los Alamos, New Mexico (1965)