Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 5, p. 897-921
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.
DOI : https://doi.org/10.1051/m2an:2006038
Classification:  35Q30,  65N35
Keywords: Stokes problem; vorticity, velocity and pressure formulation; spectral element methods
@article{M2AN_2006__40_5_897_0,
     author = {Amoura, Karima and Bernardi, Christine and Chorfi, Nejmeddine},
     title = {Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {5},
     year = {2006},
     pages = {897-921},
     doi = {10.1051/m2an:2006038},
     zbl = {1109.76044},
     mrnumber = {2293251},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_5_897_0}
}
Amoura, Karima; Bernardi, Christine; Chorfi, Nejmeddine. Spectral element discretization of the vorticity, velocity and pressure formulation of the Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 5, pp. 897-921. doi : 10.1051/m2an:2006038. http://www.numdam.org/item/M2AN_2006__40_5_897_0/

[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. | Zbl pre02132407

[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. | Zbl 0914.35094

[3] F. Ben Belgacem and C. Bernardi, Spectral element discretization of the Maxwell equations. Math. Comput. 68 (1999) 1497-1520. | Zbl 0932.65110

[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 Analysis V, P.G. Ciarlet and J.-L. Lions Eds., North-Holland (1997) 209-485. | Zbl 1117.65159

[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. | Zbl 0521.76027

[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. | Zbl 0976.46023

[9] A. Buffa, M. Costabel and M. Dauge, Algebraic convergence for anisotropic edge elements in polyhedral domains. Numer. Math. 101 (2005) 29-65. | Zbl 1116.78020

[10] M. Costabel and M. Dauge, Espaces fonctionnels Maxwell: Les gentils, les méchants et les singularités, Web publication (1998) http://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. | Zbl 1116.78002

[12] F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091-1119. | Zbl 1099.76049

[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. | Zbl 1070.76014

[14] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | MR 851383 | Zbl 0585.65077

[15] J.-C. Nédélec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[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. | Zbl 0362.65089

[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).