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On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. (English) Zbl 0692.76002
M. J. Crochet and J. M. Marchal [(*) J. Non-Newtonian Fluid Mech. 26, 77-114 (1987; Zbl 0637.76009)] introduced a new mixed finite element method for the numerical simulation of viscoelastic flows. One of the properties of this method is that it gives a good approximation (i.e., of optimal order) of the limiting newtonian case, although a nonclassical one. This paper presents a proof of this result, using the theory of mixed finite element methods as developed by e.g.: F. Brezzi [Revue Franc. Autom. Inform. Rech. Operat. 8, R-2, 129-151 (1974; Zbl 0338.90047)]. Its outline goes as follows. In the first section we quickly review the main facts about the approximation of mixed variational problems, in the second we present the problem itself, while in the third we study the discretization of Crochet and Marchal (*).

MSC:
76A10 Viscoelastic fluids
76M99 Basic methods in fluid mechanics
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[1] Brezzi, F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO anal. numér., 8, 129-151, (1974), (R.2) · Zbl 0338.90047
[2] F. Brezzi and M. Fortin, in preparation.
[3] Ciarlet, P.G.; Raviart, P.A., Interpolation theory over curved element, with applications to finite element methods, Comput. methods. appl. mech. engrg., 1, 217-249, (1972) · Zbl 0261.65079
[4] Crochet, M.J.; Marchal, J.M., A new mixed finite element for calculating viscoelastic flow, J. non-Newtonian fluid mech., 26, 77-114, (1987) · Zbl 0637.76009
[5] Fortin, M.; Fortin, A., A new approach for the FEM simulation of viscoelastic flows, Jnnfm, (1989), to appear. · Zbl 0672.76010
[6] Fortin, M., An analysis of the convergence of mixed finite element methods, RAIRO anal. numér., 341-354, (1977), (R2) · Zbl 0373.65055
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