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

Pingbing Ming

doi:10.1051/m2an:2000115

All issues

Volume 34 / No 5 (September/October 2000)

ESAIM: M2AN, 34 5 (2000) 1051-1067

References

Free Access

Issue		ESAIM: M2AN Volume 34, Number 5, September/October 2000


Page(s)		1051 - 1067
DOI		https://doi.org/10.1051/m2an:2000115
Published online		15 April 2002

R.A. Adams, Sobolev Space. Academic Press, New York (1975). [Google Scholar]
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). [Google Scholar]
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. [Google Scholar]
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. [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods, SIAM J. Numer. Anal. 28 (1991) 581-590. [Google Scholar]
F. Brezzi and M. Fortin, Mixed and Hybrid Methods. Springer-Verlags, New York (1991). [Google Scholar]
P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1978). [Google Scholar]
M.J. Crochet, A.R. Davis and K. Walters, Numerical Simulations of Non-Newtonian Flow. Elsevier, Amsterdam, Rheology Series 1 (1984). [Google Scholar]
M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations. RAIRO Anal. Numér. 3 (1973) 33-75. [Google Scholar]
M. Fortin, Old and new finite elements for incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 347-364. [CrossRef] [MathSciNet] [Google Scholar]
M. Fortin, R. Guénette and R. Pierre, Numerical analysis of the modified EVSS method. Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95. [CrossRef] [MathSciNet] [Google Scholar]
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. [CrossRef] [MathSciNet] [Google Scholar]
V. Girault and R.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin-New York (1986). [Google Scholar]
P. Hood and C. Taylor, A numerical solution of the Navier-Stokes equation using the finite element technique. Comput and Fluids 1 (1973) 73-100. [CrossRef] [MathSciNet] [Google Scholar]
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. [Google Scholar]
J. Malek and S.J. Necas, Weak and Measure-valued Solution to Evolutionary Partial Differential Equations. Chapman & Hall (1996). [Google Scholar]
Pingbing Ming and Zhong-ci Shi, Dual combined finite element methods for Non-Newtonian flow (I) Nonlinear Stabilized Methods (1998 Preprint) [Google Scholar]
Pingbing Ming and Zhong-ci Shi, A technique for the analysis of B-B inequality for non-Newtonian flow (1998 Preprint). [Google Scholar]
D. Sandri, Analyse d'une formulation à trois champs du problème de Stokes. RAIRO Modél. Math. Math. Anal. Numér. 27 (1993) 817-841. [Google Scholar]
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. [MathSciNet] [Google Scholar]
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. [CrossRef] [Google Scholar]
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). [Google Scholar]
B. Szabó and I. Babuska, Finite Element Analysis. John & Sons, Inc. (1991). [Google Scholar]
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. [Google Scholar]

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[1] R.A. Adams, Sobolev Space. Academic Press, New York (1975). [Google Scholar]

[2] 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). [Google Scholar]

[3] 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. [Google Scholar]

[4] 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. [Google Scholar]

[5] 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. [CrossRef] [MathSciNet] [Google Scholar]

[6] F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods, SIAM J. Numer. Anal. 28 (1991) 581-590. [Google Scholar]

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Methods. Springer-Verlags, New York (1991). [Google Scholar]

[8] P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1978). [Google Scholar]

[9] M.J. Crochet, A.R. Davis and K. Walters, Numerical Simulations of Non-Newtonian Flow. Elsevier, Amsterdam, Rheology Series 1 (1984). [Google Scholar]

[10] M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations. RAIRO Anal. Numér. 3 (1973) 33-75. [Google Scholar]

[11] M. Fortin, Old and new finite elements for incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 347-364. [CrossRef] [MathSciNet] [Google Scholar]

[12] M. Fortin, R. Guénette and R. Pierre, Numerical analysis of the modified EVSS method. Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95. [CrossRef] [MathSciNet] [Google Scholar]

[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. [CrossRef] [MathSciNet] [Google Scholar]

[14] V. Girault and R.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin-New York (1986). [Google Scholar]

[15] P. Hood and C. Taylor, A numerical solution of the Navier-Stokes equation using the finite element technique. Comput and Fluids 1 (1973) 73-100. [CrossRef] [MathSciNet] [Google Scholar]

[16] 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. [Google Scholar]

[17] J. Malek and S.J. Necas, Weak and Measure-valued Solution to Evolutionary Partial Differential Equations. Chapman & Hall (1996). [Google Scholar]

[18] Pingbing Ming and Zhong-ci Shi, Dual combined finite element methods for Non-Newtonian flow (I) Nonlinear Stabilized Methods (1998 Preprint) [Google Scholar]

[19] Pingbing Ming and Zhong-ci Shi, A technique for the analysis of B-B inequality for non-Newtonian flow (1998 Preprint). [Google Scholar]

[20] D. Sandri, Analyse d'une formulation à trois champs du problème de Stokes. RAIRO Modél. Math. Math. Anal. Numér. 27 (1993) 817-841. [Google Scholar]

[21] 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. [MathSciNet] [Google Scholar]

[22] 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. [CrossRef] [Google Scholar]

[23] 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). [Google Scholar]

[24] B. Szabó and I. Babuska, Finite Element Analysis. John & Sons, Inc. (1991). [Google Scholar]

[25] 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. [Google Scholar]