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Some new families of finite elements for the Stokes equations. (English) Zbl 0708.76088

Summary: We introduce a way of using the mixed finite element families of P. A. Raviart and J. M. Thomas [Math. Aspects Finite Elem. Mech., Proc. Conf. Rome 1975, Lect. Notes Math. 606, 292-315 (1977; Zbl 0362.65089)] and J.-C. Nedelec [Numer. Math. 35, 315-341 (1980; Zbl 0436.65087)], and F. Brezzi et al. [e.g. with J. Douglas, R. Durán, M. Fortin, ibid. 51, 237-250 (1987; Zbl 0631.65107)], for constructing stable and optimally convergent discretizations for the Stokes problem.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:

[1] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO. M2AN,19, 7-32 (1985) · Zbl 0567.65078
[2] Babu?ka, I., Osborn, J.E., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039-1062 (1980) · Zbl 0472.65083
[3] Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Ser. Rouge8, 129-151 (1974) · Zbl 0338.90047
[4] Brezzi, F., Douglas, J.: Stabilized mixed methods for Stokes problem. Numer. Math.53, 225-236 (1988) · Zbl 0669.76052
[5] Brezzi, F., Douglas, J., Durán, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math.51, 237-250 (1987) · Zbl 0631.65107
[6] Brezzi, F., Douglas, J., Fortin, M., Marini, L.D.: Efficient rectangular mixed finite elements in two and three space variables. RAIRO. M2AN21, 181-204 (1987) · Zbl 0689.65065
[7] Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic equations. Numer. Math.47, 19-34 (1985) · Zbl 0599.65072
[8] Brezzi, F. Marini, L.D., Pietra, P.: Two-dimensional exponential fitting and applications to semiconductor device equations. SIAM J. Num. Anal. (to appear) · Zbl 0686.65088
[9] Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations for the Stokes problem. In: Hackbusch, W. (ed.) Notes on numerical fluid mechanics, Vol. 10, pp. 11-19. Braunschweig: Vieweg 1984
[10] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[11] Douglas, J., Santos, J.E.: Approximation of waves in composite media, the mathematics of finite elements and applications VI. MAFELAP 1987. In: Whiteman, J.R. (ed.), pp. 55-74. London: Academic Press 1988
[12] Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. Heidelberg Berlin New York: Springer 1986 · Zbl 0585.65077
[13] Nedelec, J.C.: Mixed finite elements in ?3. Numer. Math.35, 315-341 (1980) · Zbl 0419.65069
[14] Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Proceedings of the Symposium on the Mathematical Aspects of the Finite Element Method. Lect. Notes Math.606, 292-315, Springer 1977 · Zbl 0362.65089
[15] Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Number. Math.48, 447-462 (1986) · Zbl 0563.65072
[16] Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math.53, 513-538 (1988) · Zbl 0632.73063
[17] Stenberg, R.: Postprocessing schemes for some mixed finite elements. Rapport de recherche No. 800, INRIA Fevier 1988. SIAM J. Num. Anal. (to appear)
[18] Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numer.18, 175-182 (1984) · Zbl 0557.76037
[19] Wheeler, M., Gonzales, R.: Mixed finite element methods for petroleum reservoir simulation. In: Glowinski, R., Lions, J.L. (eds.) Computing methods in applied sciences and engineering, VI, pp. 639-658. Amsterdam: North-Holland 1984
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