Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions


Authors: M. Farhloul, S. Nicaise and L. Paquet
Journal: Math. Comp. 69 (2000), 965-986
MSC (1991): Primary 65N30
Published electronically: March 3, 2000
MathSciNet review: 1681112
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

This paper is concerned with the mixed formulation of the Boussinesq equations in two-dimensional domains and its numerical approximation. The paper deals first with existence and uniqueness results, as well as the description of the regularity of any solution. The problem is then approximated by a mixed finite element method, where the gradient of the velocity and the gradient of the temperature, quantities of practical importance, are introduced as new unknowns. An existence result for the finite element solution and convergence results are proved near a nonsingular solution. Quasi-optimal error estimates are finally presented.


References [Enhancements On Off] (What's this?)

  • 1. Christine Bernardi, Brigitte Métivet, and Bernadette Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: le modèle et son approximation par éléments finis, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 7, 871–921 (French, with English and French summaries). MR 1364404
  • 2. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
  • 3. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • 4. Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439
  • 5. Monique Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489, 10.1137/0520006
  • 6. M. Farhloul, Métdes d'éléments finis mixtes et des volumes finis, Thèse, Université Laval, Québec, 1991.
  • 7. M. Farhloul and H. Manouzi, Analysis of non-singular solutions of a mixed Navier-Stokes formulation, Comput. Methods Appl. Mech. Engrg. 129 (1996), no. 1-2, 115–131. MR 1376913, 10.1016/0045-7825(95)00908-6
  • 8. M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal. 30 (1993), no. 4, 971–990. MR 1231323, 10.1137/0730051
  • 9. M. Farhloul and H. Manouzi, Analysis of non-singular solutions of a mixed Navier-Stokes formulation, Comput. Methods Appl. Mech. Engrg. 129 (1996), no. 1-2, 115–131. MR 1376913, 10.1016/0045-7825(95)00908-6
  • 10. M. Farhloul, S. Nicaise and L. Paquet, A mixed formulation of Boussinesq equations: Analysis of non-singular solutions, Math. Comp. (to appear). CMP 99:10
  • 11. J.-M. Ghidaglia, Etude d'écoulements de fluides incompressibles: comportement pour les grands temps et applications aux attracteurs, Thèse, Université de Paris-Sud, Centre d'Orsay, France, 1984.
  • 12. Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • 13. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • 14. Luc Paquet, The Boussinesq equations in the presence of thermocapillarity at some part of the boundary, Partial differential equations (Han-sur-Lesse, 1993) Math. Res., vol. 82, Akademie-Verlag, Berlin, 1994, pp. 266–278. MR 1322753
  • 15. P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
  • 16. N. Rouche and J. Mawhin, Équations différentielles ordinaires, Masson et Cie, Éditeurs, Paris, 1973 (French). Tome II: Stabilité et solutions périodiques. MR 0481182

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N30

Retrieve articles in all journals with MSC (1991): 65N30


Additional Information

M. Farhloul
Affiliation: Université de Moncton, Département de Mathématiques et de Statistique, N.B., E1A 3 E9, Moncton, Canada
Email: farhlom@umoncton.ca

S. Nicaise
Affiliation: Université de Valenciennes et du Hainaut Cambrésis, LIMAV, ISTV, B.P. 311, F-59304 - Valenciennes Cedex, France
Email: snicaise@univ_valenciennes.fr

L. Paquet
Affiliation: Université de Valenciennes et du Hainaut Cambrésis, LIMAV, ISTV, B.P. 311, F-59304 - Valenciennes Cedex, France
Email: Luc.Paquet@univ_valenciennes.fr

DOI: https://doi.org/10.1090/S0025-5718-00-01186-8
Received by editor(s): May 9, 1997
Received by editor(s) in revised form: July 15, 1998
Published electronically: March 3, 2000
Article copyright: © Copyright 2000 American Mathematical Society