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.
Mots clés : three-dimensional Navier-Stokes equations, mixed boundary conditions, finite element methods, a priori error estimates, a posteriori error estimates
@article{M2AN_2009__43_6_1185_0,
author = {Bernardi, Christine and Hecht, Fr\'ed\'eric and Verf\"urth, R\"udiger},
title = {A finite element discretization of the three-dimensional {Navier-Stokes} equations with mixed boundary conditions},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1185--1201},
publisher = {EDP-Sciences},
volume = {43},
number = {6},
year = {2009},
doi = {10.1051/m2an/2009035},
mrnumber = {2588437},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2009035/}
}
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: Modélisation mathématique et analyse numérique PY - 2009 SP - 1185 EP - 1201 VL - 43 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2009035/ DO - 10.1051/m2an/2009035 LA - en ID - M2AN_2009__43_6_1185_0 ER -
%0 Journal Article %A Bernardi, Christine %A Hecht, Frédéric %A Verfürth, Rüdiger %T A finite element discretization of the three-dimensional Navier-Stokes equations with mixed boundary conditions %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 1185-1201 %V 43 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2009035/ %R 10.1051/m2an/2009035 %G en %F M2AN_2009__43_6_1185_0
Bernardi, Christine; Hecht, Frédéric; Verfürth, Rüdiger. A finite element discretization of the three-dimensional Navier-Stokes equations with mixed boundary conditions. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1185-1201. doi : 10.1051/m2an/2009035. http://archive.numdam.org/articles/10.1051/m2an/2009035/
[1] , , and , Vorticity-velocity-pressure formulation for the Stokes problem. Math. Comput. 73 (2003) 1673-1697. | Zbl
[2] , and , Stabilized finite element method for the Navier-Stokes equations with physical boundary conditions. Math. Comput. 76 (2007) 1195-1217. | MR | Zbl
[3] , , and , Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823-864. | MR | Zbl
[4] , , and , 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. | Zbl
[5] , and , Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications 45. Springer (2004). | MR | Zbl
[6] and , Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer, Berlin (1991). | MR | Zbl
[7] , and , Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980/1981) 1-25. | MR | Zbl
[8] , , and , Navier-Stokes equations with imposed pressure and velocity fluxes. Internat. J. Numer. Methods Fluids 20 (1995) 267-287. | MR | Zbl
[9] , A remark on the regularity of solutions of Maxwell's equations on Lipschitz domain. Math. Meth. Appl. Sci. 12 (1990) 365-368. | MR | Zbl
[10] and , Computation of resonance frequencies for Maxwell equations in non smooth domains, in Topics in Computational Wave Propagation, Springer (2004) 125-161. | MR | Zbl
[11] , Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091-1119. | MR | Zbl
[12] , and , Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pures Appl. 82 (2003) 1395-1451. | MR | Zbl
[13] , Differential forms on Riemannian manifolds. Comm. Pure Appl. Math. 8 (1955) 551-590. | MR | Zbl
[14] and , Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer (1986). | MR | Zbl
[15] , , and , Freefem++. Second edition, v. 3.0-1, Université Pierre et Marie Curie, Paris, France (2007), http://www.freefem.org/ff++/ftp/freefem++doc.pdf.
[16] , Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Mathématiques & Applications 13. Springer (1993). | MR | Zbl
[17] and , Problèmes aux limites non homogènes et applications 1. Dunod (1968). | Zbl
[18] and , 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. | MR | Zbl
[19] and , Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213-231. | MR | Zbl
[20] , 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] , Basic Linear Partial Differential Equations. Academic Press (1975). | MR | Zbl
[22] , A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley (1996). | Zbl
Cité par Sources :
