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Dual-mixed finite element methods for the Navier-Stokes equations∗∗∗

Published online by Cambridge University Press:  29 March 2013

Jason S. Howell  and
Noel J. Walkington
Jason S. Howell
Affiliation:
Department of Mathematics, College of Charleston, Charleston, 29424 SC, USA. howelljs@cofc.edu
Noel J. Walkington
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, 15213 PA, USA; noelw@andrew.cmu.edu
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Abstract

A mixed finite element method for the Navier–Stokes equations is introduced in which thestress is a primary variable. The variational formulation retains the mathematicalstructure of the Navier–Stokes equations and the classical theory extends naturally tothis setting. Finite element spaces satisfying the associated inf–sup conditions aredeveloped.

Keywords

Navier–Stokes equationsmixed methods

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 789 - 805
Copyright
© EDP Sciences, SMAI, 2013

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References

Arnold, D.N., Brezzi, F. and Douglas, J. Jr., PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. Google Scholar
Arnold, D.N., Douglas, J. Jr. and Gupta, C.P., A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45 (1984) 1–22. Google Scholar
Arnold, D.N., Falk, R.S. and Winther, R., Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76 (2007) 1699–1723 (electronic). Google Scholar
Barlow, J., Optimal stress location in finite element method. Internat. J. Numer. Methods Engrg. 10 (1976) 243–251. Google Scholar
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.
Boffi, D., Brezzi, F. and Fortin, M., Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8 (2009) 95–121. Google Scholar
Brezzi, F., Douglas, J. Jr. and Marini, L.D., Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. Google Scholar
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Comput. Math. Springer-Verlag, New York 15 (1991).
Brezzi, F., Rappaz, J. and Raviart, P.-A., Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980/81) 1–25.
Cai, Z., Wang, C. and Zhang, S., Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. SIAM J. Numer. Anal. 48 (2010) 79–94. Google Scholar
Cai, Z. and Wang, Y., Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 63 (2010) 341–356. Google Scholar
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. Google Scholar
Cockburn, B., Gopalakrishnan, J. and Guzmán, J., A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79 (2010) 1331–1349. Google Scholar
Farhloul, M. and Manouzi, H., Analysis of non-singular solutions of a mixed Navier-Stokes formulation. Comput. Methods Appl. Mech. Engrg. 129 (1996) 115–131. Google Scholar
Farhloul, M., Nicaise, S. and Paquet, L., A refined mixed finite-element method for the stationary Navier-Stokes equations with mixed boundary conditions. IMA J. Numer. Anal. 28 (2008) 25–45. Google Scholar
Farhloul, M., Nicaise, S. and Paquet, L., 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. Google Scholar
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).
Gopalakrishnan, J. and Guzmán, J., A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012) 352–372.Google Scholar
Howell, J.S. and Walkington, N.J., Inf-sup conditions for twofold saddle point problems. Numer. Math. 118 (2011) 663–693. Google Scholar
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).
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.
Scott, L.R. and Vogelius, M., 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. Google Scholar
Shapiro, A., 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. Google Scholar
R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI 49 (1997).
Stenberg, R., Analysis of mixed finite elements methods for the Stokes problem: a unified approach. Math. Comput. 42 (1984) 9–23. Google Scholar
Stenberg, R., A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513–538. Google Scholar
R. Temam, Navier-Stokes Equations, North Holland (1977).
Zhang, S., A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74 (2005) 543–554. Google Scholar
Zhang, Z., Ultraconvergence of the patch recovery technique. Math. Comput. 65 (1996) 1431–1437. Google Scholar
Zienkiewicz, O.C., Taylor, R. and Too, J., Reduced integration technique in general analysis of plates and shells. Inter. J. Numer. Methods Engrg. 3 (1971) 275–290. Google Scholar
Zienkiewicz, O.C. and Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates I. The recovery technique. Internat. J. Numer. Methods Engrg. 33 (1992) 1331–1364. Google Scholar
Zienkiewicz, O.C. and Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates. II. Error estimates and adaptivity. Inter. J. Numer. Methods Engrg. 33 (1992) 1365–1382. Google Scholar
Zienkiewicz, O.C. and Zhu, J.Z., 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). Google Scholar