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.
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {38},
year = {2004},
pages = {781-810},
doi = {10.1051/m2an:2004039},
mrnumber = {2104429},
zbl = {1079.76055},
language = {en},
url = {http://dml.mathdoc.fr/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 781-810. doi : 10.1051/m2an:2004039. http://gdmltest.u-ga.fr/item/M2AN_2004__38_5_781_0/
[1] , Méthodes spectrales et d'éléments spectraux pour les équations de Navier-Stokes axisymétriques. Thesis, Université Pierre et Marie Curie, Paris (1997).
[2] , A mixed stream function and vorticity formulation for axisymmetric Navier-Stokes equations. J. Comp. Appl. Math. 117 (2000) 61-83. | Zbl 0966.76063
[3] , and, Une formulation mixte convergente pour le système de Stokes tridimensionnel. C. R. Acad. Sci. Paris Série I 328 (1999) 935-938. | Zbl 0945.76042
[4] , and, Une formulation mixte convergente pour les équations de Stokes tridimensionnelles2001) 61-68. | Zbl 1079.65550
[5] ,, and, Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823-864. | Zbl 0914.35094
[6] and, Spectral element discretization of the Maxwell equations. Math. Comput. 68 (1999) 1497-1520. | Zbl 0932.65110
[7] and, Approximations spectrales de problèmes aux limites elliptiques. Springer-Verlag. Math. Appl. 10 (1992). | MR 1208043 | Zbl 0773.47032
[8] , and, 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 0753.76130
[9] , and, 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 0830.46015
[10] ,, and, Spectral Methods for Axisymmetric Domains. Gauthier-Villars & North-Holland. Ser. Appl. Math. 3 (1999). | MR 1693480 | Zbl 0929.35001
[11] and, Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38 (1982) 67-86. | Zbl 0567.41008
[12] and, Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221-276. | Zbl 0968.35113
[13] , Analyse numérique des problèmes d'écoulement de fluides. Thesis, Université de Pau et des Pays de l'Adour, Pau (2001).
[14] and, An analysis of a mixed finite element method for the Navier-Stokes equations. Numer. Math. 33 (1979) 235-271. | Zbl 0396.65070
[15] and, Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | Zbl 0585.65077
[16] and, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review 21 (1979) 167-212. | Zbl 0427.65073