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

Nehla Abdellatif; Christine Bernardi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 5, page 781-810
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Abdellatif, Nehla, and 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 38.5 (2004): 781-810. <http://eudml.org/doc/245019>.

@article{Abdellatif2004,
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.},
author = {Abdellatif, Nehla, Bernardi, Christine},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Stokes problem; spectral methods; axisymmetric geometries; Fourier expansion; vector potential; optimal error estimates},
language = {eng},
number = {5},
pages = {781-810},
publisher = {EDP-Sciences},
title = {A new formulation of the Stokes problem in a cylinder, and its spectral discretization},
url = {http://eudml.org/doc/245019},
volume = {38},
year = {2004},
}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 5
SP - 781
EP - 810
AB - 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.
LA - eng
KW - Stokes problem; spectral methods; axisymmetric geometries; Fourier expansion; vector potential; optimal error estimates
UR - http://eudml.org/doc/245019
ER -

References

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  1. [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. [2] N. Abdellatif, A mixed stream function and vorticity formulation for axisymmetric Navier–Stokes equations. J. Comp. Appl. Math. 117 (2000) 61–83. Zbl0966.76063
  3. [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. Zbl0945.76042
  4. [4] M. Amara, H. Barucq and M. Duloué, Une formulation mixte convergente pour les équations de Stokes tridimensionnelles. Actes des VI es Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques, Publ. Univ. Pau, Pau (2001) 61–68. Zbl1079.65550
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  9. [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. Zbl0830.46015
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  12. [12] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221–276. Zbl0968.35113
  13. [13] M. Duloué, Analyse numérique des problèmes d’écoulement de fluides. Thesis, Université de Pau et des Pays de l’Adour, Pau (2001). 
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