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A new formulation of the Stokes problem in a cylinder, and its spectral discretization

Published online by Cambridge University Press:  15 October 2004

Nehla Abdellatif  and
Christine Bernardi
Nehla Abdellatif
Affiliation:
École Nationale des Sciences de l'Informatique, Campus Universitaire, 2010 Manouba, Tunisia.
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, CNRS & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. [email protected].
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Abstract

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

Keywords

Stokes problemspectral methodsaxisymmetric geometries.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 5 , September 2004 , pp. 781 - 810
Copyright
© EDP Sciences, SMAI, 2004

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References

N. Abdellatif, Méthodes spectrales et d'éléments spectraux pour les équations de Navier–Stokes axisymétriques. Thesis, Université Pierre et Marie Curie, Paris (1997).
Abdellatif, N., A mixed stream function and vorticity formulation for axisymmetric Navier–Stokes equations. J. Comp. Appl. Math. 117 (2000) 61–83. CrossRef
Amara, M., Barucq, H. and Duloué, M., Une formulation mixte convergente pour le système de Stokes tridimensionnel. C. R. Acad. Sci. Paris Série I 328 (1999) 935–938. CrossRef
M. Amara, H. Barucq and M. Duloué, Une formulation mixte convergente pour les équations de Stokes tridimensionnelles. Actes des VIes Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques, Publ. Univ. Pau, Pau (2001) 61–68.
Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional nonsmooth domains. Math. Meth. 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
C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer-Verlag. Math. Appl. 10 (1992).
Bernardi, C., Girault, V. and Maday, Y., Mixed spectral element approximation of the Navier-Stokes equations in the stream-function and vorticity formulation. IMA J. Numer. Anal. 12 (1992) 565–608. CrossRef
Bernardi, C., Dauge, M. and Maday, Y., Interpolation of nullspaces for polynomial approximation of divergence-free functions in a cube, in Proc. Conf. Boundary Value Problems and Integral Equations in Non smooth Domains, M. Costabel, M. Dauge and S. Nicaise Eds., Dekker. Lect. Notes Pure Appl. Math. 167 (1994) 27–46.
C. Bernardi, M. Dauge, Y. Maday and M. Azaïez, Spectral Methods for Axisymmetric Domains. Gauthier-Villars & North-Holland. Ser. Appl. Math. 3 (1999).
Canuto, C. and Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38 (1982) 67–86. CrossRef
Costabel, M. and Dauge, M., Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221–276. CrossRef
M. Duloué, Analyse numérique des problèmes d'écoulement de fluides. Thesis, Université de Pau et des Pays de l'Adour, Pau (2001).
Girault, V. and Raviart, P.-A., An analysis of a mixed finite element method for the Navier-Stokes equations. Numer. Math. 33 (1979) 235–271. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for the Navier–Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).
Glowinski, R. and Pironneau, O., Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review 21 (1979) 167–212. CrossRef