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Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows

Published online by Cambridge University Press:  15 November 2006

Andrea Bonito ,
Philippe Clément  and
Marco Picasso
Andrea Bonito
Affiliation:
Institut d'Analyse et Calcul Scientifique, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. andrea.bonito@a3.epfl.ch; marco.picasso@epfl.ch
Philippe Clément
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, NL-2300 RA Leiden, The Netherlands. PPJEClement@netscape.net
Marco Picasso
Affiliation:
Institut d'Analyse et Calcul Scientifique, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. andrea.bonito@a3.epfl.ch; marco.picasso@epfl.ch
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Abstract

A simplified stochastic Hookean dumbbells model arising from viscoelastic flows is considered, the convective terms being disregarded. A finite element discretization in space is proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using an implicit function theorem and regularity results obtained in [Bonito et al., J. Evol. Equ.6 (2006) 381–398] for the solution of the continuous problem. A posteriori error estimates are also derived. Numerical results with small time steps and a large number of realizations confirm the convergence rate with respect to the mesh size.

Keywords

ViscoelasticHookean dumbbellsfinite elementsstochastic differential equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 4 , July 2006 , pp. 785 - 814
Copyright
© EDP Sciences, SMAI, 2006

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References

Akhavan, R. and Zhou, Q., A comparison of FENE and FENE-P dumbbell and chain models in turbulent flow. J. Non-Newton. Fluid 109 (2003) 115–155.
Arada, N. and Sequeira, A., Strong steady solutions for a generalized Oldroyd-B model with shear-dependent viscosity in a bounded domain. Math. Mod. Meth. Appl. S. 13 (2003) 1303–1323. CrossRef
Baaijens, F.P.T., Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newton. Fluid 79 (1998) 361–385. CrossRef
Babuška, I., Durán, R. and Rodríguez, 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) 931–947.
Baranger, J. and Sandri, D., Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. I. Discontinuous constraints. Numer. Math. 63 (1992) 13–27. CrossRef
Baranger, J. and Wardi, S., Numerical analysis of a FEM for a transient viscoelastic flow. Comput. Method. Appl. M. 125 (1995) 171–185. CrossRef
Barrett, J.W., Schwab, C. and Süli, E., Existence of global weak solutions for some polymeric flow models. Math. Mod. Meth. Appl. S. 15 (2005) 939–983. CrossRef
R. Bird, C. Curtiss, R. Armstrong and O. Hassager, Dynamics of polymeric liquids, Vol. 1 and 2. John Wiley & Sons, New York, 1987.
A. Bonito, Ph. Clément and M. Picasso, Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math. (submitted).
Bonito, A., Clément, Ph. and Picasso, M., Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows. J. Evol. Equ. 6 (2006) 381–398. CrossRef
Bonito, A., Picasso, M. and Laso, M., Numerical simulation of 3d viscoelastic flows with complex free surfaces. J. Comput. Phys. 215 (2006) 691–716. CrossRef
Bonvin, J. and Picasso, M., Variance reduction methods for connffessit-like simulations. J. Non-Newton. Fluid 84 (1999) 191–215. CrossRef
Bonvin, J. and Picasso, M., A finite element/Monte-Carlo method for polymer dilute solutions. Comput. Vis. Sci. 4 (2001) 93–98. Second AMIF International Conference (Il Ciocco, 2000). CrossRef
Bonvin, J., Picasso, M. and Stenberg, R., EVSS, GLS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Method. Appl. M. 190 (2001) 3893–3914. CrossRef
van den Brule, B.H.A.A., van Heel, A.P.G. and Hulsen, M.A., Simulation of viscoelastic flows using Brownian configuration fields. J. Non-Newton. Fluid 70 (1997) 79–101.
van den Brule, B.H.A.A., van Heel, A.P.G. and Hulsen, M.A., On the selection of parameters in the FENE-P model. J. Non-Newton. Fluid 75 (1998) 253–271.
van den Brule, B.H.A.A, Hulsen, M.A. and Öttinger, H.C., Brownian configuration fields and variance reduced connffessit. J. Non-Newton. Fluid 70 (1997) 255–261.
B. Buffoni and J. Toland, Analytic theory of global bifurcation. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, (2003).
G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems, in Handbook of numerical analysis, Vol. V, North-Holland, Amsterdam (1997) 487–637.
Chauvière, C. and Lozinski, A., A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model. J. Comput. Phys. 189 (2003) 607–625.
P.G. Ciarlet and J.-L. Lions, editors, Handbook of numerical analysis. Vol. II. North-Holland, Amsterdam, (1991). Finite element methods. Part 1.
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84.
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992).
E, W., Li, T. and Zhang, P., Convergence of a stochastic method for the modeling of polymeric fluids. Acta Math. Sin. 18 (2002) 529–536.
E, W., Li, T. and Zhang, P., Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys. 248 (2004) 409–427. CrossRef
Ervin, V.J. and Heuer, N., Approximation of time-dependent, viscoelastic fluid flow: Crank-Nicolson, finite element approximation. Numer. Methods Partial Differ. Equ. 20 (2004) 248–283. CrossRef
Ervin, V.J. and Miles, W.W., Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal. 41 (2003) 457–486 (electronic). CrossRef
Fan, X., Molecular models and flow calculation: I. the numerical solutions to multibead-rod models in inhomogeneous flows. Acta Mech. Sin. 5 (1989) 49–59.
Fan, X., Molecular models and flow calculation: II. simulation of steady planar flow. Acta Mech. Sin. 5 (1989) 216–226.
Farhloul, M. and Zine, A.M., A new mixed finite element method for viscoelastic fluid flows. Int. J. Pure Appl. Math. 7 (2003) 93–115.
E. Fernández-Cara, F. Guillén and R.R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in Handbook of numerical analysis, Vol. VIII, North-Holland, Amsterdam (2002) 543–661.
Fortin, A., Guénette, R. and Pierre, R., On the discrete EVSS method. Comput. Method. Appl. M. 189 (2000) 121–139. CrossRef
Fortin, M. and Pierre, R., On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Method. Appl. M. 73 (1989) 341–350. CrossRef
Giga, Y., Analyticity of the semigroup generated by the Stokes operator in $L\sb{r}$ spaces. Math. Z. 178 (1981) 297–329. CrossRef
Grande, E., Laso, M. and Picasso, M., Calculation of variable-topology free surface flows using CONNFFESSIT. J. Non-Newton. Fluid 113 (2003) 127–145. CrossRef
Guillopé, C. and Saut, J.-C., Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal-theor. 15 (1990) 849–869. CrossRef
Jourdain, B., Lelièvre, T. and Le Bris, C., Numerical analysis of micro-macro simulations of polymeric fluid flows: a simple case. Math. Mod. Meth. Appl. S. 12 (2002) 1205–1243. CrossRef
Jourdain, B., Le Bris, C. and Lelièvre, T., On a variance reduction technique for micro-macro simulations of polymeric fluids. J. Non-Newton. Fluid 122 (2004) 91–106. CrossRef
Jourdain, B., Lelièvre, T. and Le Bris, C., Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209 (2004) 162–193. CrossRef
Jourdain, B., Le Bris, C., Lelièvre, T. and Otto, F., Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. An. 181 (2006) 97–148. CrossRef
I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York (1991).
Keunings, R., On the Peterlin approximation for finitely extensible dumbbells. J. Non-Newton. Fluid 68 (1997) 85–100. CrossRef
R. Keunings, Micro-marco methods for the multi-scale simulation of viscoelastic flow using molecular models of kinetic theory, in Rheology Reviews, D.M. Binding, K. Walters (Eds.), British Society of Rheology (2004) 67–98.
Larsson, S., Thomée, V. and Wahlbin, L.B., Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comp. 67 (1998) 45–71. CrossRef
Laso, M. and Öttinger, H.C., Calculation of viscoelastic flow using molecular models: the connffessit approach. J. Non-Newton. Fluid 47 (1993) 1–20. CrossRef
Laso, M., Öttinger, H.C. and Picasso, M., 2-d time-dependent viscoelastic flow calculations using connffessit. AICHE Journal 43 (1997) 877–892. CrossRef
Le Bris, C. and Lions, P.-L., Renormalized solutions of some transport equations with partially W 1,1 velocities and applications. Ann. Mat. Pur. Appl. 183 (2004) 97–130. CrossRef
Lelièvre, T., Optimal error estimate for the CONNFFESSIT approach in a simple case. Comput. Fluids 33 (2004) 815–820. CrossRef
Li, T. and Zhang, P., Convergence analysis of BCF method for Hookean dumbbell model with finite difference scheme. Multiscale Model. Simul. 5 (2006) 205–234. CrossRef
Li, T., Zhang, H. and Zhang, P., Local existence for the dumbbell model of polymeric fluids. Comm. Partial Diff. Eq. 29 (2004) 903–923. CrossRef
Lions, P.L. and Masmoudi, N., Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B 21 (2000) 131–146. CrossRef
Lozinski, A. and Owens, R.G., An energy estimate for the Oldroyd B model: theory and applications. J. Non-Newton. Fluid 112 (2003) 161–176. CrossRef
A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16 Birkhäuser Verlag, Basel (1995).
Machmoum, A. and Esselaoui, D., Finite element approximation of viscoelastic fluid flow using characteristics method. Comput. Method. Appl. M. 190 (2001) 5603–5618. 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
H.C. Öttinger, Stochastic processes in polymeric fluids. Springer-Verlag, Berlin (1996).
R.G. Owens and T.N. Phillips, Computational rheology. Imperial College Press, London (2002).
Picasso, M. and Rappaz, J., Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows. ESAIM: M2AN 35 (2001) 879–897. CrossRef
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Number 23 in Springer Series in Computational Mathematics. Springer-Verlag (1991).
Renardy, M., Existence of slow steady flows of viscoelastic fluids of integral type. Z. Angew. Math. Mech. 68 (1988) T40–T44.
Renardy, M., An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22 (1991) 313–327. CrossRef
D. Revuz and M. Yor, Continuous martingales and Brownian motion, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 293 Springer-Verlag, Berlin (1994).
Sandri, D., Analyse d'une formulation à trois champs du problème de Stokes. RAIRO Modél. Math. Anal. Numér. 27 (1993) 817–841. CrossRef
Sandri, D., Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. Continuous approximation of the stress. SIAM J. Numer. Anal. 31 (1994) 362–377. CrossRef
Sobolevskiĭ, P.E., Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR 157 (1964) 52–55.
Verfürth, R., A posteriori error estimators for the Stokes equations. Numer. Math. 55 (1989) 309–325. CrossRef
von Petersdorff, T. and Schwab, Ch., Numerical solution of parabolic equations in high dimensions. ESAIM: M2AN 38 (2004) 93–127. CrossRef
Zhang, H. and Zhang, P., Local existence for the FENE-Dumbbells model of polymeric liquids. Arch. Ration. Mech. An. 181 (2006) 373–400. CrossRef