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Existence, a priori and a posteriori error estimatesfor a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Published online by Cambridge University Press:  15 April 2002

Marco Picasso  and
Jacques Rappaz
Marco Picasso
Affiliation:
Département de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
Jacques Rappaz
Affiliation:
Département de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
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Abstract

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B modelwithout convective terms is considered. Assuming the domain to be a convexpolygon, existence of a solutionis proved for small relaxation times.Continuous piecewise linear finite elements together witha Galerkin Least Square (GLS) method are studied for solving this problem.Existence and a priori error estimatesare established using a Newton-chord fixed point theorem,a posteriori error estimates are also derived.An Elastic Viscous Split Stress (EVSS) scheme related to the GLS methodis introduced. Numerical results confirm the theoretical predictions.

Keywords

Viscoelastic fluidsGalerkin Least Square finite elements.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 5 , September 2001 , pp. 879 - 897
Copyright
© EDP Sciences, SMAI, 2001

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References

Baaijens, F.P.T., Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech. 79 (1998) 361-385. CrossRef
Babuska, I., Duran, R., and Rodriguez, R., Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal. 29 (1992) 947-964. CrossRef
Baranger, J. and El-Amri, H., Estimateurs a posteriori d'erreur pour le calcul adaptatif d'écoulements quasi-newtoniens. RAIRO Modél. Math. Anal. Numér. 25 (1991) 31-48. CrossRef
Baranger, J. and Sandri, D., Finite element approximation of viscoelastic fluid flow. Numer. Math. 63 (1992) 13-27. CrossRef
Behr, M., Franca, L., and Tezduyar, T., Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows. Comput. Methods Appl. Mech. Engrg. 104 (1993) 31-48. CrossRef
Bonvin, J., Picasso, M. and Stenberg, R., EVSS, GLS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Methods Appl. Mech. Engrg. 190 (2001) 3893-3914. CrossRef
J.C. Bonvin, Numerical simulation of viscoelastic fluids with mesoscopic models. Ph.D. thesis, Département de Mathématiques, École Polytechnique Fédérale de Lausanne (2000).
Bonvin, J.C. and Picasso, M., Variance reduction methods for CONNFFESSIT-like simulations. J. Non-Newtonian Fluid Mech. 84 (1999) 191-215. CrossRef
G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems, in Handbook of Numerical Analysis. Vol. V: Techniques of Scientific Computing (Part 2), P.G. Ciarlet and J.L. Lions, Eds., Elsevier, Amsterdam (1997) 487-637.
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
Clément, P., Approximation by finite elements using local regularization. RAIRO Anal. Numér. 8 (1975) 77-84.
Fortin, M., Guénette, R., and Pierre, R., Numerical analysis of the modified EVSS method. Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95. CrossRef
Fortin, M. and Pierre, R., On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Methods Appl. Mech. Engrg. 73 (1989) 341-350. CrossRef
Franca, L., Frey, S., and Hughes, T.J.R., Stabilized finite element methods: Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg. 95 (1992) 253-276. CrossRef
Franca, L. and Stenberg, R., Error analysis of some GLS methods for elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680-1697. CrossRef
X. Gallez, P. Halin, G. Lielens, R. Keunings, and V. Legat, The adaptative Lagrangian particle method for macroscopic and micro-macro computations of time-dependent viscoelastic flows. Comput. Methods Appl. Mech. Engrg. 180 (199) 345-364.
Girault, V. and Scott, L.R., Analysis of a 2nd grade-two fluid model with a tangential boundary condition. J. Math. Pures Appl. 78 (1999) 981-1011. CrossRef
P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985).
Guillopé, C. and Saut, J.-C., Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15 (1990) 849-869. CrossRef
Hulsen, M.A., van Heel, A.P.G., and van den Brule, B.H.A.A., Simulation of viscoelastic clows using Brownian configuration Fields. J. Non-Newtonian Fluid Mech. 70 (1997) 79-101. CrossRef
Najib, K. and Sandri, D., On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow. Numer. Math. 72 (1995) 223-238. CrossRef
Quinzani, L.M., Armstrong, R.C., and Brown, R.A., Birefringence and Laser-Doppler velocimetry studies of viscoelastic flow through a planar contraction. J. Non-Newtonian Fluid Mech. 52 (1994) 1-36. CrossRef
Renardy, M., Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z. Angew. Math. Mech. 65 (1985) 449-451. CrossRef
Ruas, V., Finite element methods for the three-field stokes system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 489-525. CrossRef
Sandri, D., Analysis of a three-fields approximation of the stokes problem. RAIRO Modél. Math. Anal. Numér. 27 (1993) 817-841. CrossRef
Sequeira, A. and Baia, M., A finite element approximation for the steady solution of a second-grade fluid model. J. Comput. Appl. Math. 111 (1999) 281-295. CrossRef
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam, New York, Oxford (1984).
Verfürth, R., A posteriori error estimators for the Stokes equations. Numer. Math. 55 (1989) 309-325. CrossRef