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Stabilized mixed methods for the Stokes problem. (English) Zbl 0669.76052
The solution of the Stokes problem is approximated by three stabilized mixed methods, one due to Th. J. R. Hughes, M. Balestra, and L. P. Franca [Comput. Methods Appl. Mech. Eng. 59, 85-99 (1986; Zbl 0622.76077)] and the other two being variants of this procedure. In each case the bilinear form associated with the saddle-point problem of the standard mixed formulation is modified to become coercive over the finite element space. Error estimates are derived for each procedure.

MSC:
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] Brezzi, F., Pitkäranta, J.: On the stabilization of Finite Element Approximations of the Stokes Problem. In: Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, Vol. 10 (W. Hackbusch, ed.), pp. 11-19. Braunschweig, Wiesbaden: Viewig 1984
[2] Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Berlin Heidelberg New York: Springer 1986 · Zbl 0585.65077
[3] Hughes, T.J.R., Franca, L.P., Balestra, M.: A New Finite Element Formulation for Computational Fluid Mechanics: V. Circumventing the Babu?ka-Brezzi condition: A Stable Petrov-Galerkin Formulation of the Stokes Problem Accommodating Equal Order Interpolation. Comput. Methods Appl. Mech. Eng.59, 85-99 (1986) · Zbl 0622.76077 · doi:10.1016/0045-7825(86)90025-3
[4] Nitsche, J.A.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamburg36, 9-15 (1970/1971) · Zbl 0229.65079 · doi:10.1007/BF02995904
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