First vorticity-velocity-pressure numerical scheme for the Stokes problem

François Dubois , Michel Salaün , Stéphanie Salmon (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.

Mots clés: Stokes problem; Vorticity–velocity–pressure formulation; Stream function-vorticity formulation; Mixed finite elements method; Inf–sup conditions

Domaines: Equations aux dérivées partielles [math.AP]

François Dubois, Michel Salaün, Stéphanie Salmon. First vorticity-velocity-pressure numerical scheme for the Stokes problem. Computer Methods in Applied Mechanics and Engineering, 2003, 192, num. 44-46, pp.4877-4907. ⟨10.1016/S0045-7825(03)00377-3⟩. ⟨hal-00096376⟩