Skip to main content Accessibility help
×
Home

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

  • Jason S. Howell (a1) and Noel J. Walkington (a2)

Abstract

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.

Copyright

References

Hide All
[1] Arnold, D.N., Brezzi, F. and Douglas, J. Jr., PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347367.
[2] 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) 122.
[3] Arnold, D.N., Falk, R.S. and Winther, R., Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76 (2007) 16991723 (electronic).
[4] Barlow, J., Optimal stress location in finite element method. Internat. J. Numer. Methods Engrg. 10 (1976) 243251.
[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.
[6] Boffi, D., Brezzi, F. and Fortin, M., Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8 (2009) 95121.
[7] Brezzi, F., Douglas, J. Jr. and Marini, L.D., Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217235.
[8] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Comput. Math. Springer-Verlag, New York 15 (1991).
[9] Brezzi, F., Rappaz, J. and Raviart, P.-A., Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980/81) 125.
[10] Cai, Z., Wang, C. and Zhang, S., Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. SIAM J. Numer. Anal. 48 (2010) 7994.
[11] Cai, Z. and Wang, Y., Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 63 (2010) 341356.
[12] Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 7784.
[13] Cockburn, B., Gopalakrishnan, J. and Guzmán, J., A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79 (2010) 13311349.
[14] Farhloul, M. and Manouzi, H., Analysis of non-singular solutions of a mixed Navier-Stokes formulation. Comput. Methods Appl. Mech. Engrg. 129 (1996) 115131.
[15] 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) 2545.
[16] 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) 843869.
[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).
[18] Gopalakrishnan, J. and Guzmán, J., A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012) 352372.
[19] Howell, J.S. and Walkington, N.J., Inf-sup conditions for twofold saddle point problems. Numer. Math. 118 (2011) 663693.
[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).
[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.
[22] 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) 111143.
[23] 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) 24202425.
[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).
[25] Stenberg, R., Analysis of mixed finite elements methods for the Stokes problem: a unified approach. Math. Comput. 42 (1984) 923.
[26] Stenberg, R., A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513538.
[27] R. Temam, Navier-Stokes Equations, North Holland (1977).
[28] Zhang, S., A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74 (2005) 543554.
[29] Zhang, Z., Ultraconvergence of the patch recovery technique. Math. Comput. 65 (1996) 14311437.
[30] 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) 275290.
[31] 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) 13311364.
[32] 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) 13651382.
[33] Zienkiewicz, O.C. and Zhu, J.Z., The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Engrg. 101 (1992) 207224. Reliability in computational mechanics (Kraków 1991).

Keywords

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

  • Jason S. Howell (a1) and Noel J. Walkington (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed