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

Christine Bernardi; Frédéric Hecht; Rüdiger Verfürth

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 6, page 1185-1201
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Bernardi, Christine, Hecht, Frédéric, and Verfürth, Rüdiger. "A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis 43.6 (2009): 1185-1201. <http://eudml.org/doc/250614>.

@article{Bernardi2009,
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. },
author = {Bernardi, Christine, Hecht, Frédéric, Verfürth, Rüdiger},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Three-dimensional Navier–Stokes equations; mixed boundary conditions; finite element methods; a priori error estimates; a posteriori error estimates.; three-dimensional Navier-Stokes equations; a priori error estimates; a posteriori error estimates},
language = {eng},
month = {8},
number = {6},
pages = {1185-1201},
publisher = {EDP Sciences},
title = {A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions},
url = {http://eudml.org/doc/250614},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Bernardi, Christine
AU - Hecht, Frédéric
AU - Verfürth, Rüdiger
TI - A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/8//
PB - EDP Sciences
VL - 43
IS - 6
SP - 1185
EP - 1201
AB - 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.
LA - eng
KW - Three-dimensional Navier–Stokes equations; mixed boundary conditions; finite element methods; a priori error estimates; a posteriori error estimates.; three-dimensional Navier-Stokes equations; a priori error estimates; a posteriori error estimates
UR - http://eudml.org/doc/250614
ER -

References

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  1. M. Amara, D. Capatina-Papaghiuc, E. Chacón-Vera and D. Trujillo, Vorticity-velocity-pressure formulation for the Stokes problem. Math. Comput.73 (2003) 1673–1697.  Zbl1299.76059
  2. M. Amara, D. Capatina-Papaghiuc and D. Trujillo, Stabilized finite element method for the Navier-Stokes equations with physical boundary conditions. Math. Comput.76 (2007) 1195–1217.  Zbl1119.76037
  3. C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci.21 (1998) 823–864.  Zbl0914.35094
  4. 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 SeminarIX, H. Brezis and J.-L. Lions Eds., Pitman (1988) 179–264.  Zbl0687.35069
  5. C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications45. Springer (2004).  
  6. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics15. Springer, Berlin (1991).  Zbl0788.73002
  7. 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.  Zbl0488.65021
  8. C. Conca, C. Parés, O. Pironneau and M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes. Internat. J. Numer. Methods Fluids20 (1995) 267–287.  Zbl0831.76011
  9. M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domain. Math. Meth. Appl. Sci.12 (1990) 365–368.  
  10. 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.  Zbl1116.78002
  11. F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci.25 (2002) 1091–1119.  Zbl1099.76049
  12. F. Dubois, M. Salaün and S. Salmon, Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pures Appl.82 (2003) 1395–1451.  Zbl1070.76014
  13. K.O. Friedrichs, Differential forms on Riemannian manifolds. Comm. Pure Appl. Math.8 (1955) 551–590.  Zbl0066.07504
  14. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer (1986).  Zbl0585.65077
  15. 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), .  URIhttp://www.freefem.org/ff++/ftp/freefem++doc.pdf
  16. O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Mathématiques & Applications13. Springer (1993).  Zbl0797.58005
  17. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications1. Dunod (1968).  Zbl0165.10801
  18. 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.  Zbl0826.35095
  19. J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math.69 (1994) 213–231.  Zbl0822.65034
  20. 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).  
  21. F. Trèves, Basic Linear Partial Differential Equations. Academic Press (1975).  Zbl0305.35001
  22. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley (1996).  Zbl0853.65108

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