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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 5, page 1051-1067
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topMing, Pingbing, and Zhong-ci Shi. "Dual Combined Finite Element Methods For Non-Newtonian Flow (II) Parameter-Dependent Problem." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 1051-1067. <http://eudml.org/doc/197612>.

@article{Ming2010,

abstract = {
This is the second part of the paper for a Non-Newtonian flow. Dual
combined Finite Element Methods are used to investigate the little
parameter-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 some
stable 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.
},

author = {Ming, Pingbing, Zhong-ci Shi},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Dual combined FEM; non-Newtonian flow; parameter-independent error bounds.},

language = {eng},

month = {3},

number = {5},

pages = {1051-1067},

publisher = {EDP Sciences},

title = {Dual Combined Finite Element Methods For Non-Newtonian Flow (II) Parameter-Dependent Problem},

url = {http://eudml.org/doc/197612},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Ming, Pingbing

AU - Zhong-ci Shi

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 5

SP - 1051

EP - 1067

AB -
This is the second part of the paper for a Non-Newtonian flow. Dual
combined Finite Element Methods are used to investigate the little
parameter-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 some
stable 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.

LA - eng

KW - Dual combined FEM; non-Newtonian flow; parameter-independent error bounds.

UR - http://eudml.org/doc/197612

ER -

## References

top- 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. Zbl0817.73058
- 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. Zbl0844.76004
- J.W. Barrett and W.B. Liu, Quasi-norm error bounds for the finite element approximation of a Non-Newtonian flow. Numer. Math. 61 (1994) 437-456. Zbl0811.76036
- F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods, SIAM J. Numer. Anal. 28 (1991) 581-590. Zbl0731.76042
- F. Brezzi and M. Fortin, Mixed and Hybrid Methods. Springer-Verlags, New York (1991). Zbl0788.73002
- P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1978). Zbl0383.65058
- M.J. Crochet, A.R. Davis and K. Walters, Numerical Simulations of Non-Newtonian Flow. Elsevier, Amsterdam, Rheology Series1 (1984).
- M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations. RAIRO Anal. Numér.3 (1973) 33-75. Zbl0302.65087
- M. Fortin, Old and new finite elements for incompressible flows. Internat. J. Numer. Methods Fluids1 (1981) 347-364. Zbl0467.76030
- 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
- 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
- V. Girault and R.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin-New York (1986). Zbl0585.65077
- P. Hood and C. Taylor, A numerical solution of the Navier-Stokes equation using the finite element technique. Comput and Fluids1 (1973) 73-100. Zbl0328.76020
- A.F.D. Loula and J.W.C. Guerreiro, Finite element analysis of nonlinear creeping flows. Comput. Methods Appl. Mech. Engrg.79 (1990) 89-109. Zbl0716.73091
- 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) Zbl1072.76567
- Pingbing Ming and Zhong-ci Shi, A technique for the analysis of B-B inequality for non-Newtonian flow (1998 Preprint). Zbl1072.76568
- D. Sandri, Analyse d'une formulation à trois champs du problème de Stokes. RAIRO Modél. Math. Math. Anal. Numér.27 (1993) 817-841. Zbl0791.76008
- D. Sandri, 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.
- D. Sandri, A posteriori estimators for mixed finite element approximation of a fluid obeying the power law. Comput. Meths. Appl. Mech. Engrg.166 (1998) 329-340. Zbl0953.76057
- 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). Zbl0924.76052
- B. Szabó and I. Babuska, Finite Element Analysis. John & Sons, Inc. (1991). Zbl0792.73003
- 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.

## NotesEmbed ?

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