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Finite element approximation of viscoelastic fluid flow using characteristics method. (English) Zbl 1012.76047
Summary: It is known that for numerical approximation of Oldroyd’s B model for viscoelastic fluid flows some upwinding is needed for the convection of extra-stress tensor. In this paper we make numerical analysis of such an approximation with upwinding by the method of characteristics in a finite element context. The approximate stress, velocity, and pressure are, respectively, $$P_1$$ discontinuous, $$P_2$$ continuous, and $$P_1$$ continuous. 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 give an error bound.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76A10 Viscoelastic fluids
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##### References:
  Baranger, J.; Machmoum, A., A natural norm for the method of characteristics, M^2an, 33, 6, 1223-1240, (1999) · Zbl 0948.65094  Baranger, J.; Machmoum, A., Exitence of approximate solutions and error bounds for viscoelastic fluid flow: characteristics method, Comput. meth. appl. mech. eng., 148, 39-52, (1997) · Zbl 0923.76098  Baranger, J.; Esselaoui, D.; Machmoum, A., Error estimate for convection problem with characteristics method, Numer. algorithms, 21, 49-56, (1999) · Zbl 0954.76039  Baranger, J.; Sandri, D., Finite element approximation of viscoelastic fluid flow: existence of solutions and error bounds. I-discontinuous contraints, Numer. math., 63, 13-27, (1992) · Zbl 0761.76032  Bermudez, A.; Durany, J., La méthode des caractéristiques pour LES problèmes de convection – diffusion stationnaires, M^2an, 21, 1, 7-26, (1987) · Zbl 0613.65121  Girault, V.; Raviart, P.A., Finite element method for navier – stokes equations, theory and algorithms, (1986), Springer Berlin · Zbl 0396.65070  Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043  Pironneau, O., On the transport-diffusion algorithm and its application to the navier – stokes equations, Numer. math., 38, 309, (1982) · Zbl 0505.76100  Keunings, R., On the high weissenberg number problem, J. non-Newtonian fluid mech., 20, 209-226, (1986) · Zbl 0589.76021  Marchal, J.M.; Crochet, M.J., A new finite element for calculating viscoelastic flow, J. non-Newtonian fluid mech., 26, 449-451, (1987) · Zbl 0637.76009  Fortin, M.; Fortin, A., Une note sur LES méthodes de caractéristiques et de lesaint-Raviart pour LES problèmes hyperboliques stationnaires, M^2an, 23, 4, 593-596, (1989) · Zbl 0687.65088  Fortin, M.; Esselaoui, D., A finite element procedure for viscoelastic flows, Int. J. numer. meth., 7, (1987) · Zbl 0634.76007  Basombrio, F.G., Flows in viscoelastic fluids treated by the method of characteristics, J. non-Newtonian fluid mech., 39, 17-34, (1991) · Zbl 0718.76015  Sandri, D., Remarques sur une formulation à trois champs du problème de Stokes equations, Numer. math., 27, 817-841, (1993) · Zbl 0791.76008
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