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

Andrea Bonito; Philippe Clément; Marco Picasso

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 4, page 785-814
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Bonito, Andrea, Clément, Philippe, and Picasso, Marco. "Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows." ESAIM: Mathematical Modelling and Numerical Analysis 40.4 (2006): 785-814. <http://eudml.org/doc/249762>.

@article{Bonito2006,
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. },
author = {Bonito, Andrea, Clément, Philippe, Picasso, Marco},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Viscoelastic; Hookean dumbbells; finite elements; stochastic differential equations.; viscoelastic; stochastic differential equations},
language = {eng},
month = {11},
number = {4},
pages = {785-814},
publisher = {EDP Sciences},
title = {Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows},
url = {http://eudml.org/doc/249762},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Bonito, Andrea
AU - Clément, Philippe
AU - Picasso, Marco
TI - Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/11//
PB - EDP Sciences
VL - 40
IS - 4
SP - 785
EP - 814
AB - 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.
LA - eng
KW - Viscoelastic; Hookean dumbbells; finite elements; stochastic differential equations.; viscoelastic; stochastic differential equations
UR - http://eudml.org/doc/249762
ER -

References

top
  1. R. Akhavan and Q. Zhou, A comparison of FENE and FENE-P dumbbell and chain models in turbulent flow. J. Non-Newton. Fluid109 (2003) 115–155.  
  2. N. Arada and A. Sequeira, 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.  
  3. F.P.T. Baaijens, Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newton. Fluid79 (1998) 361–385.  
  4. I. Babuška, R. Durán and R. Rodríguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal.29 (1992) 947–964.  
  5. 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) 931–947.  
  6. J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. I. Discontinuous constraints. Numer. Math.63 (1992) 13–27.  
  7. J. Baranger and S. Wardi, Numerical analysis of a FEM for a transient viscoelastic flow. Comput. Method. Appl. M.125 (1995) 171–185.  
  8. J.W. Barrett, C. Schwab and E. Süli, Existence of global weak solutions for some polymeric flow models. Math. Mod. Meth. Appl. S.15 (2005) 939–983.  
  9. R. Bird, C. Curtiss, R. Armstrong and O. Hassager, Dynamics of polymeric liquids, Vol. 1 and 2. John Wiley & Sons, New York, 1987.  
  10. A. Bonito, Ph. Clément and M. Picasso, Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math. (submitted).  
  11. A. Bonito, Ph. Clément and M. Picasso, Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows. J. Evol. Equ.6 (2006) 381–398.  
  12. A. Bonito, M. Picasso and M. Laso, Numerical simulation of 3d viscoelastic flows with complex free surfaces. J. Comput. Phys.215 (2006) 691–716.  
  13. J. Bonvin and M. Picasso, Variance reduction methods for connffessit-like simulations. J. Non-Newton. Fluid84 (1999) 191–215.  
  14. J. Bonvin and M. Picasso, A finite element/Monte-Carlo method for polymer dilute solutions. Comput. Vis. Sci.4 (2001) 93–98. Second AMIF International Conference (Il Ciocco, 2000).  
  15. J. Bonvin, M. Picasso and R. Stenberg, GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Method. Appl. M.190 (2001) 3893–3914.  
  16. B.H.A.A. van den Brule, A.P.G. van Heel and M.A. Hulsen, Simulation of viscoelastic flows using Brownian configuration fields. J. Non-Newton. Fluid70 (1997) 79–101.  
  17. B.H.A.A. van den Brule, A.P.G. van Heel and M.A. Hulsen, On the selection of parameters in the FENE-P model. J. Non-Newton. Fluid75 (1998) 253–271.  
  18. B.H.A.A van den Brule, M.A. Hulsen and H.C. Öttinger, Brownian configuration fields and variance reduced connffessit. J. Non-Newton. Fluid70 (1997) 255–261.  
  19. B. Buffoni and J. Toland, Analytic theory of global bifurcation. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, (2003).  
  20. 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.  
  21. C. Chauvière and A. Lozinski, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model. J. Comput. Phys.189 (2003) 607–625.  
  22. P.G. Ciarlet and J.-L. Lions, editors, Handbook of numerical analysis. Vol. II. North-Holland, Amsterdam, (1991). Finite element methods. Part 1.  
  23. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.9 (1975) 77–84.  
  24. 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).  
  25. W. E, T. Li and P. Zhang, Convergence of a stochastic method for the modeling of polymeric fluids. Acta Math. Sin.18 (2002) 529–536.  
  26. W. E, T. Li and P. Zhang, Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys.248 (2004) 409–427.  
  27. V.J. Ervin and N. Heuer, Approximation of time-dependent, viscoelastic fluid flow: Crank-Nicolson, finite element approximation. Numer. Methods Partial Differ. Equ.20 (2004) 248–283.  
  28. V.J. Ervin and W.W. Miles, Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal.41 (2003) 457–486 (electronic).  
  29. X. Fan, Molecular models and flow calculation: I. the numerical solutions to multibead-rod models in inhomogeneous flows. Acta Mech. Sin.5 (1989) 49–59.  
  30. X. Fan, Molecular models and flow calculation: II. simulation of steady planar flow. Acta Mech. Sin.5 (1989) 216–226.  
  31. M. Farhloul and A.M. Zine, A new mixed finite element method for viscoelastic fluid flows. Int. J. Pure Appl. Math.7 (2003) 93–115.  
  32. 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.  
  33. A. Fortin, R. Guénette and R. Pierre, On the discrete EVSS method. Comput. Method. Appl. M.189 (2000) 121–139.  
  34. M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows. Comput. Method. Appl. M.73 (1989) 341–350.  
  35. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z.178 (1981) 297–329.  
  36. E. Grande, M. Laso and M. Picasso, Calculation of variable-topology free surface flows using CONNFFESSIT. J. Non-Newton. Fluid113 (2003) 127–145.  
  37. C. Guillopé and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal-theor.15 (1990) 849–869.  
  38. B. Jourdain, T. Lelièvre and C. Le Bris, Numerical analysis of micro-macro simulations of polymeric fluid flows: a simple case. Math. Mod. Meth. Appl. S.12 (2002) 1205–1243.  
  39. B. Jourdain, C. Le Bris and T. Lelièvre, On a variance reduction technique for micro-macro simulations of polymeric fluids. J. Non-Newton. Fluid122 (2004) 91–106.  
  40. B. Jourdain, T. Lelièvre and C. Le Bris, Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal.209 (2004) 162–193.  
  41. B. Jourdain, C. Le Bris, T. Lelièvre and F. Otto, Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. An.181 (2006) 97–148.  
  42. I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York (1991).  
  43. R. Keunings, On the Peterlin approximation for finitely extensible dumbbells. J. Non-Newton. Fluid68 (1997) 85–100.  
  44. 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.  
  45. S. Larsson, V. Thomée and L.B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comp.67 (1998) 45–71.  
  46. M. Laso and H.C. Öttinger, Calculation of viscoelastic flow using molecular models: the connffessit approach. J. Non-Newton. Fluid47 (1993) 1–20.  
  47. M. Laso, H.C. Öttinger and M. Picasso, 2-d time-dependent viscoelastic flow calculations using connffessit. AICHE Journal43 (1997) 877–892.  
  48. C. Le Bris and P.-L. Lions, Renormalized solutions of some transport equations with partially W1,1 velocities and applications. Ann. Mat. Pur. Appl.183 (2004) 97–130.  
  49. T. Lelièvre, Optimal error estimate for the CONNFFESSIT approach in a simple case. Comput. Fluids33 (2004) 815–820.  
  50. T. Li and P. Zhang, Convergence analysis of BCF method for Hookean dumbbell model with finite difference scheme. Multiscale Model. Simul.5 (2006) 205–234.  
  51. T. Li, H. Zhang and P. Zhang, Local existence for the dumbbell model of polymeric fluids. Comm. Partial Diff. Eq.29 (2004) 903–923.  
  52. P.L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B21 (2000) 131–146.  
  53. A. Lozinski and R.G. Owens, An energy estimate for the Oldroyd B model: theory and applications. J. Non-Newton. Fluid112 (2003) 161–176.  
  54. A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16 Birkhäuser Verlag, Basel (1995).  
  55. A. Machmoum and D. Esselaoui, Finite element approximation of viscoelastic fluid flow using characteristics method. Comput. Method. Appl. M.190 (2001) 5603–5618.  
  56. 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.  
  57. H.C. Öttinger, Stochastic processes in polymeric fluids. Springer-Verlag, Berlin (1996).  
  58. R.G. Owens and T.N. Phillips, Computational rheology. Imperial College Press, London (2002).  
  59. M. Picasso and J. Rappaz, Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows. ESAIM: M2AN35 (2001) 879–897.  
  60. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Number 23 in Springer Series in Computational Mathematics. Springer-Verlag (1991).  
  61. M. Renardy, Existence of slow steady flows of viscoelastic fluids of integral type. Z. Angew. Math. Mech.68 (1988) T40–T44.  
  62. M. Renardy, An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal.22 (1991) 313–327.  
  63. 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).  
  64. D. Sandri, Analyse d'une formulation à trois champs du problème de Stokes. RAIRO Modél. Math. Anal. Numér.27 (1993) 817–841.  
  65. D. Sandri, 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.  
  66. P.E. Sobolevskiĭ, Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR157 (1964) 52–55.  
  67. R. Verfürth, A posteriori error estimators for the Stokes equations. Numer. Math.55 (1989) 309–325.  
  68. T. von Petersdorff and Ch. Schwab, Numerical solution of parabolic equations in high dimensions. ESAIM: M2AN38 (2004) 93–127.  
  69. H. Zhang and P. Zhang, Local existence for the FENE-Dumbbells model of polymeric liquids. Arch. Ration. Mech. An.181 (2006) 373–400.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.