Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T15:30:23.429Z Has data issue: false hasContentIssue false
  • English
  • Français

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

Published online by Cambridge University Press:  21 August 2009

Christine Bernardi ,
Frédéric Hecht  and
Rüdiger Verfürth
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; hecht@ann.jussieu.fr
Frédéric Hecht
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; hecht@ann.jussieu.fr
Rüdiger Verfürth
Affiliation:
Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany. ruediger.verfuerth@ruhr-uni-bochum.de
Get access

Abstract

We consider a variational formulation of the three-dimensional Navier–Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.

Keywords

Three-dimensional Navier–Stokes equationsmixed boundary conditionsfinite element methodsa priori error estimatesa posteriori error estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1185 - 1201
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amara, M., Capatina-Papaghiuc, D., Chacón-Vera, E. and Trujillo, D., Vorticity-velocity-pressure formulation for the Stokes problem. Math. Comput. 73 (2003) 1673–1697. CrossRef
Amara, M., Capatina-Papaghiuc, D. and Trujillo, D., Stabilized finite element method for the Navier-Stokes equations with physical boundary conditions. Math. Comput. 76 (2007) 1195–1217. CrossRef
Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823–864. 3.0.CO;2-B>CrossRef
C. Bègue, C. Conca, F. Murat and O. Pironneau, Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar IX, H. Brezis and J.-L. Lions Eds., Pitman (1988) 179–264.
C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications 45. Springer (2004).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer, Berlin (1991).
F. Brezzi, J. Rappaz and P.-A. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980/1981) 1–25.
Conca, C., Parés, C., Pironneau, O. and Thiriet, M., Navier-Stokes equations with imposed pressure and velocity fluxes. Internat. J. Numer. Methods Fluids 20 (1995) 267–287. CrossRef
Costabel, M., A remark on the regularity of solutions of Maxwell's equations on Lipschitz domain. Math. Meth. Appl. Sci. 12 (1990) 365–368. CrossRef
M. Costabel and M. Dauge, Computation of resonance frequencies for Maxwell equations in non smooth domains, in Topics in Computational Wave Propagation, Springer (2004) 125–161.
Dubois, F., Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091–1119. CrossRef
Dubois, F., Salaün, M. and Salmon, S., Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pures Appl. 82 (2003) 1395–1451. CrossRef
Friedrichs, K.O., Differential forms on Riemannian manifolds. Comm. Pure Appl. Math. 8 (1955) 551–590. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer (1986).
F. Hecht, A. Le Hyaric, K. Ohtsuka and O. Pironneau, Freefem++. Second edition, v. 3.0-1, Université Pierre et Marie Curie, Paris, France (2007), http://www.freefem.org/ff++/ftp/freefem++doc.pdf.
O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Mathématiques & Applications 13. Springer (1993).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 1. Dunod (1968).
M. Orlt and A.-M. Sändig, Regularity of viscous Navier-Stokes flows in nonsmooth domains, in Proc. Conf. Boundary Value Problems and Integral Equations in Nonsmooth Domain, Dekker (1995) 185–201.
Pousin, J. and Rappaz, J., Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213–231. CrossRef
S. Salmon, Développement numérique de la formulation tourbillon-vitesse-pression pour le problème de Stokes. Ph.D. Thesis, Université Pierre et Marie Curie, Paris, France (1999).
F. Trèves, Basic Linear Partial Differential Equations. Academic Press (1975).
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley (1996).