A new formulation of the Stokes problem in a cylinder, and its spectral discretization
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 781-810.

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.

DOI : 10.1051/m2an:2004039
Classification : 65N35
Mots-clés : Stokes problem, spectral methods, axisymmetric geometries 
@article{M2AN_2004__38_5_781_0,
     author = {Abdellatif, Nehla and Bernardi, Christine},
     title = {A new formulation of the {Stokes} problem in a cylinder, and its spectral discretization},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {781--810},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {5},
     year = {2004},
     doi = {10.1051/m2an:2004039},
     mrnumber = {2104429},
     zbl = {1079.76055},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2004039/}
}
TY  - JOUR
AU  - Abdellatif, Nehla
AU  - Bernardi, Christine
TI  - A new formulation of the Stokes problem in a cylinder, and its spectral discretization
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2004
SP  - 781
EP  - 810
VL  - 38
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2004039/
DO  - 10.1051/m2an:2004039
LA  - en
ID  - M2AN_2004__38_5_781_0
ER  - 
%0 Journal Article
%A Abdellatif, Nehla
%A Bernardi, Christine
%T A new formulation of the Stokes problem in a cylinder, and its spectral discretization
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2004
%P 781-810
%V 38
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2004039/
%R 10.1051/m2an:2004039
%G en
%F M2AN_2004__38_5_781_0
Abdellatif, Nehla; Bernardi, Christine. A new formulation of the Stokes problem in a cylinder, and its spectral discretization. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 781-810. doi : 10.1051/m2an:2004039. http://archive.numdam.org/articles/10.1051/m2an:2004039/

[1] N. Abdellatif, Méthodes spectrales et d'éléments spectraux pour les équations de Navier-Stokes axisymétriques. Thesis, Université Pierre et Marie Curie, Paris (1997).

[2] N. Abdellatif, A mixed stream function and vorticity formulation for axisymmetric Navier-Stokes equations. J. Comp. Appl. Math. 117 (2000) 61-83. | Zbl

[3] M. Amara, H. Barucq and M. Duloué, Une formulation mixte convergente pour le système de Stokes tridimensionnel. C. R. Acad. Sci. Paris Série I 328 (1999) 935-938. | Zbl

[4] M. Amara, H. Barucq and M. Duloué, Une formulation mixte convergente pour les équations de Stokes tridimensionnelles2001) 61-68. | Zbl

[5] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823-864. | Zbl

[6] F. Ben Belgacem and C. Bernardi, Spectral element discretization of the Maxwell equations. Math. Comput. 68 (1999) 1497-1520. | Zbl

[7] C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer-Verlag. Math. Appl. 10 (1992). | MR | Zbl

[8] C. Bernardi, V. Girault and Y. Maday, Mixed spectral element approximation of the Navier-Stokes equations in the stream-function and vorticity formulation. IMA J. Numer. Anal. 12 (1992) 565-608. | Zbl

[9] C. Bernardi, M. Dauge and Y. Maday, 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. | Zbl

[10] C. Bernardi, M. Dauge, Y. Maday and M. Azaïez, Spectral Methods for Axisymmetric Domains. Gauthier-Villars & North-Holland. Ser. Appl. Math. 3 (1999). | MR | Zbl

[11] C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38 (1982) 67-86. | Zbl

[12] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221-276. | Zbl

[13] M. Duloué, Analyse numérique des problèmes d'écoulement de fluides. Thesis, Université de Pau et des Pays de l'Adour, Pau (2001).

[14] V. Girault and P.-A. Raviart, An analysis of a mixed finite element method for the Navier-Stokes equations. Numer. Math. 33 (1979) 235-271. | Zbl

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

[16] R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review 21 (1979) 167-212. | Zbl

Cité par Sources :