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
DOI: https://doi.org/10.1090/S0025-5718-00-01186-8
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. C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par éléments finis, M$^2$AN, RAIRO Modél. Math. Anal. Numér. 29 (1995), 871-921. MR 96k:76028
  • 2. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991. MR 92d:65187
  • 3. P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978. MR 58:25001
  • 4. M. Dauge, Elliptic boundary value problems in corner domains. Smoothness and asymptotics of solutions, L.N. in Math., 1341, Springer Verlag, 1988. MR 91a:35078
  • 5. M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations, SIAM J. Math. Anal., 20 (1989), 74-97. MR 90b:35191
  • 6. M. Farhloul, Métdes d'éléments finis mixtes et des volumes finis, Thèse, Université Laval, Québec, 1991.
  • 7. M. Farhloul, Mixed and nonconforming finite element methods for the Stokes problem, Can. Appl. Math. Quart., 3 (1995) 399-418. MR 97a:76082
  • 8. M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal., 30 (1993), 971-990. MR 94g:65123
  • 9. M. Farhloul and H. Manouzi, Analysis of non-singular solutions of a mixed Navier-Stokes formulation, Comput. Methods Appl. Mech. Engrg., 129 (1996), 115-131. MR 97a:76082
  • 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. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. MR 88b:65129
  • 13. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 21, Pitman, Boston, 1985. MR 86m:35044
  • 14. L. Paquet, The Boussinesq equations in the presence of thermocapillarity at some part of the boundary, in: G. Lumer, S. Nicaise and B.-W. Schulze eds., Partial Differential Equations, Mathematical Research, 82, Akademie Verlag, Berlin, pp. 266-278, 1994. MR 96a:35146
  • 15. P.-A. Raviart and J.-M. Thomas, A mixed finite element method for $2$-nd order elliptic problems, Lecture Notes in Mathematics, 606, Springer-Verlag, New-York, pp. 292-315, 1977. MR 58:3547
  • 16. N. Rouche and J. Mawhin, Equations différentielles ordinaires, Tome II: Stabilité et solutions périodiques, Masson, 1973. MR 58:1318b

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

American Mathematical Society