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

Marco Picasso; Jacques Rappaz[1]

  • [1] Ecole Polytechnique Federale Institute of Analysis and Scientific Computing CH-1015 Lausanne Switzerland

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 5, page 879-897
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Picasso, Marco, and 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 35.5 (2001): 879-897. <http://eudml.org/doc/194078>.

@article{Picasso2001,
abstract = {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.},
affiliation = {Ecole Polytechnique Federale Institute of Analysis and Scientific Computing CH-1015 Lausanne Switzerland},
author = {Picasso, Marco, Rappaz, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {viscoelastic fluids; Galerkin least square finite elements; simplified Oldroyd-B model; convex polygon; existence of solution; small relaxation times; continuous piecewise linear finite elements; Galerkin least square method; a priori error estimates; Newton-chord fixed point theorem; a posteriori error estimates; elastic viscous split stress scheme},
language = {eng},
number = {5},
pages = {879-897},
publisher = {EDP-Sciences},
title = {Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows},
url = {http://eudml.org/doc/194078},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Picasso, Marco
AU - Rappaz, Jacques
TI - Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 5
SP - 879
EP - 897
AB - 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.
LA - eng
KW - viscoelastic fluids; Galerkin least square finite elements; simplified Oldroyd-B model; convex polygon; existence of solution; small relaxation times; continuous piecewise linear finite elements; Galerkin least square method; a priori error estimates; Newton-chord fixed point theorem; a posteriori error estimates; elastic viscous split stress scheme
UR - http://eudml.org/doc/194078
ER -

References

top
  1. [1] F.P.T. Baaijens, Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech. 79 (1998) 361–385. Zbl0957.76024
  2. [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. Zbl0759.65069
  3. [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. Zbl0712.76068
  4. [4] J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow. Numer. Math. 63 (1992) 13–27. Zbl0761.76032
  5. [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. Zbl0771.76033
  6. [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. Zbl1014.76043
  7. [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. [8] J.C. Bonvin and M. Picasso, Variance reduction methods for CONNFFESSIT-like simulations. J. Non-Newtonian Fluid Mech. 84 (1999) 191–215. Zbl0972.76054
  9. [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. [10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). Zbl0383.65058MR520174
  11. [11] P. Clément, Approximation by finite elements using local regularization. RAIRO Anal. Numér. 8 (1975) 77–84. Zbl0368.65008
  12. [12] M. Fortin, R. Guénette, and R. Pierre, Numerical analysis of the modified EVSS method. Comput. Methods Appl. Mech. Engrg. 143 (1997) 79–95. Zbl0896.76040
  13. [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. Zbl0692.76002
  14. [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. Zbl0759.76040
  15. [15] L. Franca and R. Stenberg, Error analysis of some GLS methods for elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680–1697. Zbl0759.73055
  16. [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. Zbl0966.76076
  17. [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. Zbl0961.35116
  18. [18] P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985). Zbl0695.35060
  19. [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. Zbl0729.76006
  20. [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. [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. Zbl0838.76044
  22. [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. [23] M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z. Angew. Math. Mech. 65 (1985) 449–451. Zbl0577.76014
  24. [24] V. Ruas, Finite element methods for the three-field stokes system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 489–525. Zbl0853.76041
  25. [25] D. Sandri, Analysis of a three-fields approximation of the stokes problem. RAIRO Modél. Math. Anal. Numér. 27 (1993) 817–841. Zbl0791.76008
  26. [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. Zbl0957.76033
  27. [27] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Company, Amsterdam, New York, Oxford (1984). Zbl0426.35003MR769654
  28. [28] R. Verfürth, A posteriori error estimators for the Stokes equations. Numer. Math. 55 (1989) 309–325. Zbl0674.65092

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.