Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem

Karima Amoura; Christine Bernardi; Nejmeddine Chorfi

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 40, Issue: 5, page 897-921
  • ISSN: 0764-583X

Abstract

top
We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element 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.

How to cite

top

Amoura, Karima, Bernardi, Christine, and Chorfi, Nejmeddine. "Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis 40.5 (2007): 897-921. <http://eudml.org/doc/194340>.

@article{Amoura2007,
abstract = { We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element 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. },
author = {Amoura, Karima, Bernardi, Christine, Chorfi, Nejmeddine},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stokes problem; vorticity; velocity and pressure formulation; spectral element methods.; variational formulation; optimal error estimates},
language = {eng},
month = {1},
number = {5},
pages = {897-921},
publisher = {EDP Sciences},
title = {Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem},
url = {http://eudml.org/doc/194340},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Amoura, Karima
AU - Bernardi, Christine
AU - Chorfi, Nejmeddine
TI - Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/1//
PB - EDP Sciences
VL - 40
IS - 5
SP - 897
EP - 921
AB - We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element 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.
LA - eng
KW - Stokes problem; vorticity; velocity and pressure formulation; spectral element methods.; variational formulation; optimal error estimates
UR - http://eudml.org/doc/194340
ER -

References

top
  1. M. Amara, D. Capatina-Papaghiuc, E. Chacón-Vera and D. Trujillo, Vorticity-velocity-pressure formulation for Navier-Stokes equations. Comput. Vis. Sci.6 (2004) 47–52.  Zbl1299.76059
  2. C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Method. Appl. Sci.21 (1998) 823–864.  Zbl0914.35094
  3. F. Ben Belgacem and C. Bernardi, Spectral element discretization of the Maxwell equations. Math. Comput.68 (1999) 1497–1520.  Zbl0932.65110
  4. C. Bernardi and N. Chorfi, Spectral discretization of the vorticity, velocity and pressure formulation of the Stokes problem. SIAM J. Numer. Anal.44 (2006) 826–850. bibitemBMx C. Bernardi and Y. Maday, Spectral Methods, in the Handbook of Numerical AnalysisV, P.G. Ciarlet and J.-L. Lions Eds., North-Holland (1997) 209–485.  Zbl1117.65159
  5. C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. Internal Report, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (2003).  
  6. C. Bernardi, V. Girault and P.-A. Raviart, Incompressible Viscous Fluids and their Finite Element Discretizations, in preparation.  
  7. J. Boland and R. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal.20 (1983) 722–731.  Zbl0521.76027
  8. A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Method. Appl. Sci.24 (2001) 31–48.  Zbl0976.46023
  9. A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math.101 (2005) 29–65.  Zbl1116.78020
  10. M. Costabel and M. Dauge, Espaces fonctionnels Maxwell: Les gentils, les méchants et les singularités, Web publication (1998) .  URIhttp://perso.univ-rennes1.fr/monique.dauge
  11. M. Costabel and M. Dauge, Computation of resonance frequencies for Maxwell equations in non smooth domains, in Topics in Computational Wave Propagation, M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne Eds., Springer (2004) 125–161.  Zbl1116.78002
  12. F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci.25 (2002) 1091–1119.  Zbl1099.76049
  13. F. Dubois, M. Salaün and S. Salmon, Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pure. Appl.82 (2003) 1395–1451.  Zbl1070.76014
  14. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).  Zbl0585.65077
  15. J.-C. Nédélec, Mixed finite elements in 3 . Numer. Math.35 (1980) 315–341.  Zbl0419.65069
  16. P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes Eds., Lect. Notes Math.606, Springer-Verlag (1977) 292–315.  
  17. S. Salmon, Développement numérique de la formulation tourbillon-vitesse-pression pour le problème de Stokes. Ph.D. thesis, Université Pierre et Marie Curie, Paris (1999).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.