zbMATH — the first resource for mathematics

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.

76M10 Finite element methods applied to problems in fluid mechanics
76A10 Viscoelastic fluids
Full Text: DOI
[1] Baranger, J.; Machmoum, A., A natural norm for the method of characteristics, M^2an, 33, 6, 1223-1240, (1999) · Zbl 0948.65094
[2] 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
[3] Baranger, J.; Esselaoui, D.; Machmoum, A., Error estimate for convection problem with characteristics method, Numer. algorithms, 21, 49-56, (1999) · Zbl 0954.76039
[4] 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
[5] 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
[6] Girault, V.; Raviart, P.A., Finite element method for navier – stokes equations, theory and algorithms, (1986), Springer Berlin · Zbl 0396.65070
[7] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[8] Pironneau, O., On the transport-diffusion algorithm and its application to the navier – stokes equations, Numer. math., 38, 309, (1982) · Zbl 0505.76100
[9] Keunings, R., On the high weissenberg number problem, J. non-Newtonian fluid mech., 20, 209-226, (1986) · Zbl 0589.76021
[10] 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
[11] 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
[12] Fortin, M.; Esselaoui, D., A finite element procedure for viscoelastic flows, Int. J. numer. meth., 7, (1987) · Zbl 0634.76007
[13] 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
[14] Sandri, D., Remarques sur une formulation à trois champs du problème de Stokes equations, Numer. math., 27, 817-841, (1993) · Zbl 0791.76008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.