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.

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.

DOI : 10.1051/m2an/2009035
Classification : 65N30, 65N15, 65J15
Mots clés : three-dimensional Navier-Stokes equations, mixed boundary conditions, finite element methods, a priori error estimates, a posteriori error estimates
Bernardi, Christine  ; Hecht, Frédéric  ; Verfürth, Rüdiger 1

1 Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany.
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     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},
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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://www.numdam.org/articles/10.1051/m2an/2009035/

[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. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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 Seminar IX, H. Brezis and J.-L. Lions Eds., Pitman (1988) 179-264. | Zbl

[5] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications 45. Springer (2004). | MR | Zbl

[6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer, Berlin (1991). | MR | Zbl

[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. | MR | Zbl

[8] C. Conca, C. Parés, O. Pironneau and M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes. Internat. J. Numer. Methods Fluids 20 (1995) 267-287. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[11] F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091-1119. | MR | Zbl

[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. | MR | Zbl

[13] K.O. Friedrichs, Differential forms on Riemannian manifolds. Comm. Pure Appl. Math. 8 (1955) 551-590. | MR | Zbl

[14] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer (1986). | MR | Zbl

[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), http://www.freefem.org/ff++/ftp/freefem++doc.pdf.

[16] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Mathématiques & Applications 13. Springer (1993). | MR | Zbl

[17] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 1. Dunod (1968). | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[22] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley (1996). | Zbl

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