Abstract
We consider the Stokes problem in a three-dimensional axisymmetric bounded domain with non standard conditions which involve the normal component of the velocity and tangential component of the vorticity. We reduce the three-dimensional problem into a two-dimensional one and we write a variational formulation of it with three independent unknowns: the vorticity, the velocity and the pressure. Then we propose a discretization by spectral 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.
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Abdellatif, N.: Méthodes spectrales et d’élé ments spectraux pour les équations de Navier-Stokes axisymétriques. Thesis, Université Pierre et Marie Curie, Paris 6 (1997)
Amara, M., Capatina-Papaghiuc, D., Chacon-Vera, E., Trujillo, D.: Vorticity-velocity-pressure formulation for Navier-Stokes equations. Comput. Vis. Sci. 6, 47–52 (2004)
Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)
Azaïez, M., Ben Belgacem, F.: Propriété des espaces H(div) axisymétriques et application à l’inversion du problème de Darcy par méthodes spectrales. Rapport interne 96.15, M.I.P. Université Paul Sabatier (1996)
Azaïez, M., Bernardi, C., Dauge, M., Maday, Y.: Spectral Methods for Axisymmetric Domains. Ser. Appl. Math. vol. 3. Gauthier-Villars & North-Holland, Paris, Amsterdam (1999)
Bernardi, C., Chorfi, N.: Spectral discretization of the vorticity, velocity and pressure formulation of the Stokes problem. SIAM J. Numer. Anal. 44, 826–850 (2006)
Bernardi, C., Girault, V.: Espaces duaux des domaines des opérateurs divergence et rotationnel avec trace nulle. Internal Report, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (2003)
Bernardi, C., Maday, Y.: Basic result on spectral methods. Rapport Interne R94022, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris (1994)
Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis V, pp. 209–485. North-Holland, Amsterdam (1997)
Bernardi, C., Dauge, M., Maday, Y.: Numerical analysis and spectral methods for axisymmetric domains, part I: functional prerequisite. Rapport Interne R94008, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris 6 (1994)
Bernardi, C., Hecht, F., Pironneau, O.: Coupling Darcy and Stokes equations for porous media with cracks. Math. Model. Numer. Anal. 39, 7–35 (2005)
Dubois, F.: Vorticity-velocity-pressure formulation for the Stokes problem. Math. Methods Appl. Sci. 25, 1091–1119 (2002)
Dubois, F., Salaün, M., Salmon, S.: Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pures Appl. 82, 1395–1451 (2003)
Ern, A., Guermond, J.-L., Quartapelle, L.: Vorticity-velocity formulation for the Stokes problem in 3D. Math. Methods Appl. Sci. 22, 531–546 (1999)
Girault, V.: Incompressible finite element methods for the Navier-Stokes equations with nonstandard boundary conditions in ℝ3. Math. Comput. 51, 55–74 (1988)
Girault, V., Raviart, P.-A.: Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)
Halpern, L.: Spectral Methods in polar coordinates for the Stokes problem. Application to computation in unbounded domains. Math. Comput. 65, 507–531 (1996)
Maday, Y., Pavoni, D.: Spectral approximation of axisymmetric Stokes flow. Internal Report 92034, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie. Paris (1992)
Nédélec, J.-C.: Mixed finite elements in ℝ3. Numer. Math. 35, 315–341 (1980)
Salmon, S.: Développement numérique de la formulation tourbillon-vitesse-pression pour le problème de Stokes. Thesis, Université Pierre et Marie Curie, Paris (1999)
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The research of N. Chorfi was supported by the King Saud University, D.S.F.P. Program.
An erratum to this article can be found at http://dx.doi.org/10.1007/s10915-011-9484-4
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Abdellatif, N., Chorfi, N. & Trabelsi, S. Spectral Discretization of the Axisymmetric Vorticity, Velocity and Pressure Formulation of the Stokes Problem. J Sci Comput 47, 419–440 (2011). https://doi.org/10.1007/s10915-010-9446-2
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DOI: https://doi.org/10.1007/s10915-010-9446-2