Abstract
We consider the Navier–Stokes equations in a two- or three-dimensional domain 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, and prove the existence of a solution for this problem. Next we propose a discretization by spectral methods which relies on this formulation. In the two-dimensional case, we prove quasi-optimal error estimates for the three unknowns. We conclude with some numerical experiments.
Résumé
Nous considérons les équations de Navier–Stokes dans un domaine biou tri-dimensionnel, munies de conditions aux limites non usuelles portant sur la composante normale de la vitesse et la ou les composantes tangentielles du tourbillon. Nous écrivons une formulation variationnelle de ce problème qui comporte trois inconnues indépendantes: le tourbillon, la vitesse et la pression. Nous prouvons que ce problème admet au moins une solution. Nous proposons une discrétisation par méthodes spectrales construite à partir de cette formulation. Dans le cas bidimensionnel, nous établissons des majorations quasi-optimales de l'erreur pour les trois inconnues. Nous concluons par quelques expériences numériques.
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Azaïez, M., Bernardi, C. & Chorfi, N. Spectral discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes equations. Numer. Math. 104, 1–2 (2006). https://doi.org/10.1007/s00211-006-0684-z
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DOI: https://doi.org/10.1007/s00211-006-0684-z