×

Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. (English) Zbl 0761.76032

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.
Reviewer: D.Sandri

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76A10 Viscoelastic fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Baranger, J., El Amri, H. (1991): Estimateurs à postériori pour le calcul adaptif d’écoulements quasinewtoniens. RAIRO, Modél. Math. Anal. Numér.25, 31-48
[2] Baranger, J., Najib, K. (1990): Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau. Numer. Math.58, 35-49 · Zbl 0702.76007
[3] Baranger, J., Sandri, D. (1991): Approximation par éléments finis d’écoulements de fluides viscoélastiques: Existence de solutions approchées et majoration d’erreur. I. Contraintes discontinues. C.R. Acad. Sci. Paris, Sér. I.312, 541-544 · Zbl 0718.76010
[4] Baranger, J., Sandri, D. (1992): Formulation of Stokes’s problem and the linear elasticity equations suggested by Oldroyd for viscoelastics flows. RAIRO Modél. Math. Anal. Numér.,26, 331-345 · Zbl 0738.76002
[5] Brezzi, F. (1974): On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO, Modél. Math. Anal. Numér.8, 129-151 · Zbl 0338.90047
[6] Ciarlet, P.G. (1978): The finite element method for elliptic problems. Elsevier/North-Holland, Amsterdam New York · Zbl 0383.65058
[7] Fortin, M., Pierre, R. (1989): On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Methods Appl. Mech. Eng.73, 341-350 · Zbl 0692.76002
[8] Fortin, M., Essellaoui, D. (1987): A finite element procedure for viscoelastic flows. Int. J. Numer. Meth. Fluid.7, 1035-1052 · Zbl 0634.76007
[9] Fortin, M., Fortin, A. (1989): A new finite approach for the F.E.M. simulation of viscoelastic flows. J. Non-Newtonian Fluid Mech.32, 295-310 · Zbl 0672.76010
[10] Girault, V., Raviart, P.A. (1986): Finite element method for Navier-Stokes equations, Theory and Algorithms. Springer, Berlin Heidelberg New York · Zbl 0585.65077
[11] Guillope, C., Saut, J.-C. (1987): Existence results for the flow of viscoelastic fluids with a differential constitutive law. C.R. Acad. Sci. Paris, Sér. I.,305, 489-492
[12] Guillope, C., Saut, J-C. (1990): Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. Theory Methods Appl.15, 849-869 · Zbl 0729.76006
[13] Hakim, A. (1990): Analyse mathématique de modèles de fluides viscoélastiques de type White-Metzner. Thesis, Université Paris-Sud
[14] Johnson, C., Pitkäranta, J. (1987): An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput.46, 1-26 · Zbl 0618.65105
[15] Joseph, D.D., Renardy, M., Saut, J.-C. (1985): Hyperbolicity and change of type in the flow of viscoelastic fluids. Arch. Ration. Mech. Anal.87, 213-251 · Zbl 0572.76011
[16] Keunings, R. (1986): On the high Weissenberg Number problem. J. Non-Newtonian Fluid Mech.20, 209-226 · Zbl 0589.76021
[17] Lesaint, P., Raviart, P.A. (1974): On a finite element method for solving the neutron transport equation. In: de Boor, C., ed., Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, pp. 89-123
[18] Marchal, J.M., Crochet, M.J. (1987): A new finite element for calculating viscoelastic flow. J. Non-Newtonian Fluid Mech.26, 77-114 · Zbl 0637.76009
[19] Renardy, M. (1985): Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z.A.M.M.65, 449-451 · Zbl 0577.76014
[20] Sandri, D (1991): Approximation éléments finis d;écoulements de fluides viscoélastiques: Existence de solutions approchées et majoration d’erreur. II. Contraintes continues. C.R. Acad. Sci. Paris, Sér. I.,313, 111-114 · Zbl 0737.76048
[21] Sandri, D.: Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. II. Continuous constraints. In preparation · Zbl 0737.76048
[22] Sandri, D. (1992): Sur l’approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou le modèle de Carreau. RAIRO Modél. Math. Anal. Numér. (to appear)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.