Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T19:28:31.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Dual Combined Finite Element Methods For Non-Newtonian Flow(II) Parameter-Dependent Problem

Published online by Cambridge University Press:  15 April 2002

Pingbing Ming  and
Zhong-ci Shi
Pingbing Ming
Affiliation:
Institute of Computational Mathematics, Chinese Academy of Sciences, PO Box 2719, Beijing 100080, P. R. China. (mpb@lsec.cc.ac.cn)
Zhong-ci Shi
Affiliation:
Institute of Computational Mathematics, Chinese Academy of Sciences, PO Box 2719, Beijing 100080, P. R. China. (shi@lsec.cc.ac.cn)
Get access

Abstract

This is the second part of the paper for a Non-Newtonian flow. Dual combined Finite Element Methods are used to investigate the littleparameter-dependent problem arising in a nonliner three field version of the Stokes system for incompressible fluids, where the viscosity obeys a general law including the Carreau's law and the Power law. Certain parameter-independent error bounds are obtained which solved the problem proposed by Baranger in [4] in a unifying way. We also give somestable finite element spaces by exemplifying the abstract B-B inequality. The continuous approximation for the extra stress is achieved as a by-product of the new method.

Keywords

Dual combined FEMnon-Newtonian flowparameter-independent error bounds.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 5 , September 2000 , pp. 1051 - 1067
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev Space. Academic Press, New York (1975).
C. Amrouche and V. Girault, Propriétés fonctionnelles d'opérateurs. Application au problème de stokes en dimension qualconque. Publications du Laboratoire d'Analyse Numérique, No. 90025, Université Piere et Marie Curie, Paris, France (1990).
D.N. Arnold and F. Brezzi, Some new elements for the Reissner-Mindlin plate model, Boundary Value Problems for Partial Differential Equations, edited by C. Baiocchi and J.L. Lions. Masson, Paris (1992) 287-292.
J. Baranger, K. Najib and D. Sandri, Numerical analysis of a three-field model for a Quasi-Newtonian flow. Comput. Methods. Appl. Mech. Engrg. 109(1993) 281-292.
Barrett, J.W. and Liu, W.B., Quasi-norm error bounds for the finite element approximation of a Non-Newtonian flow. Numer. Math . 61 (1994) 437-456. CrossRef
F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods, SIAM J. Numer. Anal. 28 (1991) 581-590.
F. Brezzi and M. Fortin, Mixed and Hybrid Methods. Springer-Verlags, New York (1991).
P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1978).
M.J. Crochet, A.R. Davis and K. Walters, Numerical Simulations of Non-Newtonian Flow. Elsevier, Amsterdam, Rheology Series 1 (1984).
Crouzeix, M. and Raviart, P., Conforming and nonconforming finite element methods for solving the stationary stokes equations. RAIRO Anal. Numér. 3 (1973) 33-75.
Fortin, M., Old and new finite elements for incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 347-364. CrossRef
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
V. Girault and R.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin-New York (1986).
Hood, P. and Taylor, C., A numerical solution of the Navier-Stokes equation using the finite element technique. Comput and Fluids 1 (1973) 73-100.
Loula, A.F.D. and Guerreiro, J.W.C., Finite element analysis of nonlinear creeping flows. Comput. Methods Appl. Mech. Engrg. 79 (1990) 89-109. CrossRef
J. Malek and S.J. Necas, Weak and Measure-valued Solution to Evolutionary Partial Differential Equations. Chapman & Hall (1996).
Pingbing Ming and Zhong-ci Shi, Dual combined finite element methods for Non-Newtonian flow (I) Nonlinear Stabilized Methods (1998 Preprint)
Pingbing Ming and Zhong-ci Shi, A technique for the analysis of B-B inequality for non-Newtonian flow (1998 Preprint).
Sandri, D., Analyse d'une formulation à trois champs du problème de Stokes. RAIRO Modél. Math. Math. Anal. Numér. 27 (1993) 817-841. CrossRef
Sandri, D., Sur l'approximation numérique des écoulements quasi-newtoniens dont la viscoélastiques suit la Loi Puissance ou le modèle de Carreau. RAIRO-Modèl. Math. Anal. Numér. 27 (1993) 131-155. CrossRef
Sandri, D., A posteriori estimators for mixed finite element approximation of a fluid obeying the power law. Comput. Meths. Appl. Mech. Engrg. 166 (1998) 329-340. CrossRef
C. Schwab and M. Suri, Mixed h-p finite element methods for Stokes and non-Newtonian Flow. Research report No. 97-19, Seminar für Angewandte Mathematik, ETH Zürich (1997).
B. Szabó and I. Babuska, Finite Element Analysis. John & Sons, Inc. (1991).
Tianxiao Zhou, Stabilized finite element methods for a model parameter-dependent problem, in Proc. of the Second Conference on Numerical Methods for P.D.E, edited by Longan Ying and Benyu Guo. World Scientific, Singapore (1991) 192-194.