no. 3
Dual-mixed finite element methods for the Navier-Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 789-805.

A mixed finite element method for the Navier-Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier-Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf-sup conditions are developed.

DOI : 10.1051/m2an/2012050
Classification : 65N60, 65N12, 65M60, 65M12
Mots clés : Navier-Stokes equations, mixed methods 
@article{M2AN_2013__47_3_789_0,
     author = {Howell, Jason S. and Walkington, Noel J.},
     title = {Dual-mixed finite element methods for the {Navier-Stokes} equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {789--805},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {3},
     year = {2013},
     doi = {10.1051/m2an/2012050},
     mrnumber = {3056409},
     zbl = {1266.76029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2012050/}
}
TY  - JOUR
AU  - Howell, Jason S.
AU  - Walkington, Noel J.
TI  - Dual-mixed finite element methods for the Navier-Stokes equations
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 789
EP  - 805
VL  - 47
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2012050/
DO  - 10.1051/m2an/2012050
LA  - en
ID  - M2AN_2013__47_3_789_0
ER  - 
%0 Journal Article
%A Howell, Jason S.
%A Walkington, Noel J.
%T Dual-mixed finite element methods for the Navier-Stokes equations
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 789-805
%V 47
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2012050/
%R 10.1051/m2an/2012050
%G en
%F M2AN_2013__47_3_789_0
Howell, Jason S.; Walkington, Noel J. Dual-mixed finite element methods for the Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 789-805. doi : 10.1051/m2an/2012050. http://archive.numdam.org/articles/10.1051/m2an/2012050/

[1] D.N. Arnold, F. Brezzi and J. Douglas, Jr., PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347-367. | MR | Zbl

[2] D.N. Arnold, J. Douglas, Jr. and C.P. Gupta, A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45 (1984) 1-22. | EuDML | MR | Zbl

[3] D.N. Arnold, R.S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76 (2007) 1699-1723 (electronic). | MR | Zbl

[4] J. Barlow, Optimal stress location in finite element method. Internat. J. Numer. Methods Engrg. 10 (1976) 243-251. | Zbl

[5] D. Boffi, F. Brezzi, L.F. Demkowicz, R.G. Durán, R.S. Falk and M. Fortin, Mixed finite elements, compatibility conditions, and applications, Springer-Verlag, Berlin. Lect. Notes Math. 1939 (2008). Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26-July 1, 2006, edited by Boffi and Lucia Gastaldi. | MR | Zbl

[6] D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8 (2009) 95-121. | MR | Zbl

[7] F. Brezzi, J. Douglas, Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | EuDML | MR | Zbl

[8] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Comput. Math. Springer-Verlag, New York 15 (1991). | MR | Zbl

[9] F. Brezzi, J. Rappaz and P.-A. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980/81) 1-25. | EuDML | MR | Zbl

[10] Z. Cai, C. Wang and S. Zhang, Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. SIAM J. Numer. Anal. 48 (2010) 79-94. | MR

[11] Z. Cai and Y. Wang, Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 63 (2010) 341-356. | MR

[12] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl

[13] B. Cockburn, J. Gopalakrishnan and J. Guzmán, A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79 (2010) 1331-1349. | MR

[14] M. Farhloul and H. Manouzi, Analysis of non-singular solutions of a mixed Navier-Stokes formulation. Comput. Methods Appl. Mech. Engrg. 129 (1996) 115-131. | MR | Zbl

[15] M. Farhloul, S. Nicaise and L. Paquet, A refined mixed finite-element method for the stationary Navier-Stokes equations with mixed boundary conditions. IMA J. Numer. Anal. 28 (2008) 25-45. | MR | Zbl

[16] M. Farhloul, S. Nicaise and L. Paquet, A priori and a posteriori error estimations for the dual mixed finite element method of the Navier-Stokes problem. Numer. Methods Partial Differ. Equ. 25 (2009) 843-869. | MR | Zbl

[17] V. Girault and P.A. Raviart, Finite Element Approximation of the Navier Stokes Equations. Springer Verlag, Berlin, Heidelbert, New York. Lect. Notes Math. 749 (1979). | MR | Zbl

[18] J. Gopalakrishnan and J. Guzmán, A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012) 352-372. | MR | Zbl

[19] J.S. Howell and N.J. Walkington, Inf-sup conditions for twofold saddle point problems. Numer. Math. 118 (2011) 663-693. | MR | Zbl

[20] W. Layton, Introduction to the numerical analysis of incompressible viscous flows, Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 6 (2008). | MR | Zbl

[21] P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Springer, Berlin. Lect. Notes Math. 606 (1977) 292-315. | MR | Zbl

[22] L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111-143. | Numdam | MR | Zbl

[23] A. Shapiro, The use of an exact solution of the navier-stokes equations in a validation test of a three-dimensional non-hydrostatic numerical model. Mon. Wea. Rev. 121 (1993) 2420-2425.

[24] R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI 49 (1997). | MR | Zbl

[25] R. Stenberg, Analysis of mixed finite elements methods for the Stokes problem: a unified approach. Math. Comput. 42 (1984) 9-23. | MR | Zbl

[26] R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513-538. | MR | Zbl

[27] R. Temam, Navier-Stokes Equations, North Holland (1977). | MR | Zbl

[28] S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74 (2005) 543-554. | MR | Zbl

[29] Z. Zhang, Ultraconvergence of the patch recovery technique. Math. Comput. 65 (1996) 1431-1437. | MR | Zbl

[30] O.C. Zienkiewicz, R. Taylor and J. Too, Reduced integration technique in general analysis of plates and shells. Inter. J. Numer. Methods Engrg. 3 (1971) 275-290. | Zbl

[31] O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates I. The recovery technique. Internat. J. Numer. Methods Engrg. 33 (1992) 1331-1364. | MR | Zbl

[32] O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. II. Error estimates and adaptivity. Inter. J. Numer. Methods Engrg. 33 (1992) 1365-1382. | MR | Zbl

[33] O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Engrg. 101 (1992) 207-224. Reliability in computational mechanics (Kraków 1991). | MR | Zbl

Cité par Sources :