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

Nehla Abdellatif; Christine Bernardi

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 38.5 (2010): 781-810. <http://eudml.org/doc/194240>.

@article{Abdellatif2010,
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},
keywords = {Stokes problem; spectral methods; axisymmetric geometries.; Fourier expansion; vector potential; optimal error estimates},
language = {eng},
month = {3},
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/194240},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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/194240
ER -

References

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  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. M. Amara, H. Barucq and M. Duloué, Une formulation mixte convergente pour le système de Stokes tridimensionnel. C. R. Acad. Sci. Paris Série I328 (1999) 935–938.  Zbl0945.76042
  4. 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.  Zbl1079.65550
  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.  Zbl0914.35094
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  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
  10. C. Bernardi, M. Dauge, Y. Maday and M. Azaïez, Spectral Methods for Axisymmetric Domains. Gauthier-Villars & North-Holland. Ser. Appl. Math.3 (1999).  
  11. C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput.38 (1982) 67–86.  Zbl0567.41008
  12. M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal.151 (2000) 221–276.  Zbl0968.35113
  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.  Zbl0396.65070
  15. V. Girault and P.-A. Raviart, Finite Element Methods for the Navier–Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).  Zbl0585.65077
  16. R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Review21 (1979) 167–212.  Zbl0427.65073

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