Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, p. 879-897

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are 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 method is introduced. Numerical results confirm the theoretical predictions.

Classification:  65N30,  65N12,  76A10
Keywords: viscoelastic fluids, Galerkin least square finite elements
@article{M2AN_2001__35_5_879_0,
author = {Picasso, Marco and Rappaz, Jacques},
title = {Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {5},
year = {2001},
pages = {879-897},
zbl = {0997.76051},
mrnumber = {1866272},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_5_879_0}
}

Picasso, Marco; Rappaz, Jacques. Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, pp. 879-897. http://www.numdam.org/item/M2AN_2001__35_5_879_0/

[1] F.P.T. Baaijens, Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech. 79 (1998) 361-385. | Zbl 0957.76024

[2] I. Babuska, R. Duran, and R. Rodriguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal. 29 (1992) 947-964. | Zbl 0759.65069

[3] J. Baranger and H. El-Amri, Estimateurs a posteriori d'erreur pour le calcul adaptatif d'écoulements quasi-newtoniens. RAIRO Modél. Math. Anal. Numér. 25 (1991) 31-48. | Numdam | Zbl 0712.76068

[4] J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow. Numer. Math. 63 (1992) 13-27. | Zbl 0761.76032

[5] M. Behr, L. Franca, and T. Tezduyar, Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows. Comput. Methods Appl. Mech. Engrg. 104 (1993) 31-48. | Zbl 0771.76033

[6] J. Bonvin, M. Picasso and R. Stenberg, GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Methods Appl. Mech. Engrg. 190 (2001) 3893-3914. | Zbl 1014.76043

[7] J.C. Bonvin, Numerical simulation of viscoelastic fluids with mesoscopic models. Ph.D. thesis, Département de Mathématiques, École Polytechnique Fédérale de Lausanne (2000).

[8] J.C. Bonvin and M. Picasso, Variance reduction methods for CONNFFESSIT-like simulations. J. Non-Newtonian Fluid Mech. 84 (1999) 191-215. | Zbl 0972.76054

[9] 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.

[10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

[11] P. Clément, Approximation by finite elements using local regularization. RAIRO Anal. Numér. 8 (1975) 77-84. | Numdam | Zbl 0368.65008

[12] M. Fortin, R. Guénette, and R. Pierre, Numerical analysis of the modified EVSS method. Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95. | Zbl 0896.76040

[13] M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Methods Appl. Mech. Engrg. 73 (1989) 341-350. | Zbl 0692.76002

[14] L. Franca, S. Frey, and T.J.R. Hughes, Stabilized finite element methods: Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg. 95 (1992) 253-276. | Zbl 0759.76040

[15] L. Franca and R. Stenberg, Error analysis of some GLS methods for elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680-1697. | Zbl 0759.73055

[16] 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. | Zbl 0966.76076

[17] V. Girault and L.R. Scott, Analysis of a 2nd grade-two fluid model with a tangential boundary condition. J. Math. Pures Appl. 78 (1999) 981-1011. | Zbl 0961.35116

[18] P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985). | Zbl 0695.35060

[19] C. Guillopé and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15 (1990) 849-869. | Zbl 0729.76006

[20] M.A. Hulsen, A.P.G. Van Heel, and B.H.A.A. Van Den Brule, Simulation of viscoelastic clows using Brownian configuration Fields. J. Non-Newtonian Fluid Mech. 70 (1997) 79-101.

[21] K. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow. Numer. Math. 72 (1995) 223-238. | Zbl 0838.76044

[22] L.M. Quinzani, R.C. Armstrong, and R.A. Brown, Birefringence and Laser-Doppler velocimetry studies of viscoelastic flow through a planar contraction. J. Non-Newtonian Fluid Mech. 52 (1994) 1-36.

[23] M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z. Angew. Math. Mech. 65 (1985) 449-451. | Zbl 0577.76014

[24] V. Ruas, Finite element methods for the three-field stokes system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 489-525. | Numdam | Zbl 0853.76041

[25] D. Sandri, Analysis of a three-fields approximation of the stokes problem. RAIRO Modél. Math. Anal. Numér. 27 (1993) 817-841. | Numdam | Zbl 0791.76008

[26] A. Sequeira and M. Baia, A finite element approximation for the steady solution of a second-grade fluid model. J. Comput. Appl. Math. 111 (1999) 281-295. | Zbl 0957.76033

[27] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam, New York, Oxford (1984). | MR 769654 | Zbl 0426.35003

[28] R. Verfürth, A posteriori error estimators for the Stokes equations. Numer. Math. 55 (1989) 309-325. | Zbl 0674.65092