A new $\theta$-scheme algorithm and incompressible FEM for viscoelastic fluid flows

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1. [1] R. B. BIRD, R. C. AMSTRONG and O. HASSAGER, Dynamics of Polymeric Liquid, vol. 1, Fluid Mechanics, 2nd ed. (1987) Wiley, New York. 
2. [2] D. V. BOGER, Annu. Rev. Fluid Mech., 19 (1987) 157 : 182. 
3. [3] A. N. BROOKS and T. J. R. HUGHES, Streamline-Upwind/Petrov-Galerkin Formulation for Convection Dominated Flow with Particular Emphasis on the Incompressible Navier-Stokes Equations, Comp. Meth. in Appl. Mech. and Eng,, 32 (1982) pp. 199-259. Zbl0497.76041MR679322
4. [4] M. J. CROCHET and J. M. MARCHAL, A new mixed Finite Element for calculating Viscoelastic Flow, Journal of Non-Newtonian Fluid Mechanics, 26 (1987) pp. 77-114. Zbl0637.76009
5. [5] N. El KISSI, J. M. PIAU and B. TREMBLAY, Low Reynolds number flow visualisation of linear and branched silicones upstream of orifices dies, Journal of Non-Newtonian Fluid Mechanics (1988). 
6. [6] R. E. EVANS and K. WALTERS, Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquid, Journal of Non-Newtonian Fluid Mechanics, 20 (1986) pp. 11-29. 
7. [7] M. FORTIN and A. FORTIN, A new approach for the FEM simulation of viscoelastic flow, Journal of Non-Newtonian Fluid Mechanics, 32 (1989) pp. 295-310. Zbl0672.76010
8. [8] M. FORTIN and R. GLOWINSKI, Lagrangian Augmented Methods, (1981) North Holland. 
9. [9] M. FORTIN and R. PIERRE, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flow, (1989) Comput. Meth. in Appl. Mech. Eng. Zbl0692.76002MR1016647
10. [10] V. GIRAULT and P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749, (1979) Springer Verlag. Zbl0413.65081MR548867
11. [11] R. GLOWINSKI and J. PÉRIAUX, Numerical Methods for Nonlinear Problems in Fluid Dynamics, Proceeding of the International Seminar on Scientific Super-computer, (1987) Feb 2-6. Zbl0666.76080MR1026941
12. [12] J. B. GOODMAN and R. J. LEVEQUE, On the accuracy of stable scheme for two dimensional conservation laws, Soc. Ind. Appl. Math. Numer. anal., 25 (1988)pp. 268-284. Zbl0645.65051MR933724
13. [13] C. GUILLOPÉ and J. C. SAUT, Global existence and one-dimensional non-linear stability of shearing motions of viscoelastic fluids of Oldroyd type, Modélisation Mathématique et Analyse Numérique, 24 (1990) pp. 369-401. Zbl0701.76011MR1055305
14. [14] C. GUILLOPÉ et J. C. SAUT, Résultat d'existence pour les fluides viscoélastiques à loi de comportement de type différentiel, Compte-rendu de l'Académie des Sciences de Paris, 305, série I (1987) pp. 489-492. Zbl0624.76008MR916317
15. [15] D. D. JOSEPH, M. RENARDY and J. C. SAUT, Hyperbolicity and Change of Type in the Flow of Viscoelastic Fluids, Arch. Ration. Mech. Anal, 87 (1985) pp. 213-251. Zbl0572.76011MR768067
16. [16] R. KEUNIGS, Simulation of Viscoelastic Flow, in Fundamentals of Computer Modeling for Polymer Processing, C. L. Tucker III, Cari Hanser Verlag. 
17. [17] P. LESAINT and P. A. RAVIART, On finite element methods for solving the neutron transport equation (1974) Carl de Boor, Academic Press. 
18. [18] X. L. Luo and R. I. TANNER, A Decoupled Finite Element Streamline-Upwind Scheme for Viscoelastic Flow Problems, J. of Non-Newtonian Fluid Mechanics, 31 (1989) pp. 143-162. 
19. [19] J. G. OLDROYD, On the formulation of Rheological equation of states, Proc, Roy. Soc. London, A200 (1950) pp. 523-541. Zbl1157.76305MR35192
20. [20] D. W. PEACEMAN and H. H. RACHFORD, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1955) pp. 28-41. Zbl0067.35801MR71874
21. [21] M. RENARDY, Existence of Slow Steady Flows of Viscoelastic Fluids with Differential Constitutive Equations, Z. Angew. Math, u Mech., 65 (1985) pp. 449-451. Zbl0577.76014MR814684
22. [22] M. RENARDY, Recent advances in the mathematical theory of steady flow of viscoelastic fluids, J. of Non-Newtonian Fluid Mechanics, 9 (1988) pp. 11-24. Zbl0666.76029
23. [23] J. E. ROBERTS and J. M. THOMAS, Mixed and hybrid methods, in Handbook of Numerical Analysis, vol. 3, P. G. Ciarlet and J. L. Lions, North Holland, Amsterdam (Rapport de Recherche 737, INRIA 1987). Zbl0875.65090MR1115239
24. [24] P. SARAMITO, Simulation numérique d'écoulements de fluides viscoélastiques par éléments finis incompressibles et une méthode de directions alternées ; applications, thèse de l'INPG (1990) Grenoble. 
25. [25] J. E. WELCH, F. H. HARLOW, J. P. SHANNON and B. J. DALY, The M.A.C. method, LASL report LA3425, Los Alamos Scientific Laboratory, 1965.