Coupling Harmonic Functions-Finite Elements for Solving the Stream Function-Vorticity Stokes Problem

Toufic Abboud (1) , Michel Salaün (2) , Stéphanie Salmon (3)

1 CMAP - Centre de Mathématiques Appliquées
2 CNAM Paris - Centre d'enseignement Cnam Paris
3 IRMA - Institut de Recherche Mathématique Avancée

We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity form. The classical finite element method of degree one usually used does not allow the vorticity on the boundary of the domain to be computed satisfactorily when the meshes are unstructured and does not converge optimally. To better approach the vorticity along the boundary, we propose that harmonic functions obtained by integral representation be used. Numerical results are very satisfactory, and we prove that this new numerical scheme leads to an optimal convergence rate of order 1 for the natural norm of the vorticity and, under higher regularity assumptions, from 3/2 to 2 for the quadratic norm of the vorticity.

Mots clés: Stokes problem; Stream function-vorticity formulation; Harmonic functions; Integral representation

Domaines: Mécanique des structures [physics.class-ph]; Mécanique des structures [physics.class-ph]

Toufic Abboud, Michel Salaün, Stéphanie Salmon. Coupling Harmonic Functions-Finite Elements for Solving the Stream Function-Vorticity Stokes Problem. Numerical Methods for Partial Differential Equations, 2004, vol. 20, pp. 765-788. ⟨10.1002/num.2005⟩. ⟨hal-00815176⟩