Multi-field finite element methods with discontinuous pressures for axisymmetric incompressible flow

Two- and three-field methods are studied for solving the Stokes system in the axisymmetric case, as a linearized form of different types of fluid flow equations. Both are designed for the standard Galerkin formulation expressed in terms either of the velocity and the pressure, or of these two fields together with the extra-stress tensor, and use discontinuous pressure spaces. The first method is related to the rectangular based Q2−P1 element due to Fortin. The other one is linked to the Crouzeix–Raviart triangle. Both methods satisfy the uniform stability (inf–sup) condition relating the velocity and pressure representations, expressed in terms of the natural weighed Sobolev norms, for the system under consideration. This condition is fundamental to derive second-order convergence results for solution methods of viscous or viscoelastic incompressible flow problems based on the corresponding finite element spaces. In order to illustrate this, some numerical results using a method of this type studied by the authors are presented, in connection with the three-field formulation of the Stokes system related to viscoelastic fluids.