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Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem

Published online by Cambridge University Press:  16 January 2007

Karima Amoura ,
Christine Bernardi  and
Nejmeddine Chorfi
Karima Amoura
Affiliation:
Université Badji-Mokhtar, Faculté des Sciences, Département de Mathématiques, B.P. 12, 23000 Annaba, Algeria.
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. et Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.
Nejmeddine Chorfi
Affiliation:
Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060 Tunis, Tunisia.
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Abstract

We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element methods which relies on this formulation. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.

Keywords

Stokes problem; vorticityvelocity and pressure formulation; spectral element methods.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 5 , September 2006 , pp. 897 - 921
Copyright
© EDP Sciences, SMAI, 2007

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References

Amara, M., Capatina-Papaghiuc, D., Chacón-Vera, E. and Trujillo, D., Vorticity-velocity-pressure formulation for Navier-Stokes equations. Comput. Vis. Sci. 6 (2004) 47–52. CrossRef
Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional nonsmooth domains. Math. Method. Appl. Sci. 21 (1998) 823–864. 3.0.CO;2-B>CrossRef
Ben Belgacem, F. and Bernardi, C., Spectral element discretization of the Maxwell equations. Math. Comput. 68 (1999) 1497–1520. CrossRef
Bernardi, C. and Chorfi, N., Spectral discretization of the vorticity, velocity and pressure formulation of the Stokes problem. SIAM J. Numer. Anal. 44 (2006) 826–850. bibitemBMx C. Bernardi and Y. Maday, Spectral Methods, in the Handbook of Numerical Analysis V, P.G. Ciarlet and J.-L. Lions Eds., North-Holland (1997) 209–485. CrossRef
C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. Internal Report, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (2003).
C. Bernardi, V. Girault and P.-A. Raviart, Incompressible Viscous Fluids and their Finite Element Discretizations, in preparation.
Boland, J. and Nicolaides, R., Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722–731. CrossRef
Buffa, A. and Ciarlet, P., On, Jr. traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Method. Appl. Sci. 24 (2001) 31–48. 3.0.CO;2-X>CrossRef
Buffa, A., Costabel, M. and Dauge, M., Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29–65. CrossRef
M. Costabel and M. Dauge, Espaces fonctionnels Maxwell: Les gentils, les méchants et les singularités, Web publication (1998) http://perso.univ-rennes1.fr/monique.dauge.
M. Costabel and M. Dauge, Computation of resonance frequencies for Maxwell equations in non smooth domains, in Topics in Computational Wave Propagation, M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne Eds., Springer (2004) 125–161.
Dubois, F., Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091–1119. CrossRef
Dubois, F., Salaün, M. and Salmon, S., Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pure. Appl. 82 (2003) 1395–1451. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).
Nédélec, J.-C., Mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 35 (1980) 315–341. CrossRef
P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes Eds., Lect. Notes Math. 606, Springer-Verlag (1977) 292–315.
S. Salmon, Développement numérique de la formulation tourbillon-vitesse-pression pour le problème de Stokes. Ph.D. thesis, Université Pierre et Marie Curie, Paris (1999).