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Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. I: Discontinuous constraints. (Approximation par éléments finis d’écoulements de fluides viscoélastiques: Existence de solutions approchées et majoration d’erreur. I: Contraintes discontinues.) (French) Zbl 0718.76010
Summary: We study a finite element approximation of viscoelastic fluid flow obeying an Oldroyd B type constitutive law. The approximate stress, velocity and pressure are respectively \(P_ 1\) discontinuous, \(P_ 2\) continuous, \(P_ 1\) continuous. We use the method of Lesaint-Raviart for the convection of the extra stress tensor. We suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. We show by a fixed point method that the approximate problem has a solution and we give an error bound.

76A10 Viscoelastic fluids
76M10 Finite element methods applied to problems in fluid mechanics