Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection. (Q2767584)

The authors discuss the implementation of a pressure stabilized finite element method for the incompressible Navier-Stokes equations which can be considered as GLS method, in the sense that it intends to allow the use of equal order velocity-pressure interpolations. This is possible due to a stabilization technique based on the introduction as discrete unknown the pressure gradient projection (PGP) onto the finite element space of continuous vector fields. Then divergence of the difference between these two vectors (pressure gradient and its projection) is introduced in the continuity equation multiplied by suitable algorithmic parameters. The resulting formulation is referred to as stabilized by pressure gradient projection (SPGP) method. The presence of PGP as a new unknown in the problem makes the applicability of the method questionable, since the number of nodal unknowns is increased. The main purpose of this paper is to shown that it is not necessary to solve in a fully coupled manner for the velocity, pressure and PGP. For stationary problems, this can be done by using iterative techniques, but for transient problems the natural way to uncouple the resolution is by creating explicitly the PGP. Herein it is shown that this is not only feasible, but also very convenient in general situations. Nevertheless, in some cases the iterative coupling is also found efficient. The only purpose of the stabilization technique presented here is to stabilize the pressure. The instabilities due to the convective term when the viscosity is very small are not considered in the formulation. The pressure stabilized method is fully described, and its stability and convergence properties are summarized. Numerical results are presented. It is shown that even though the computational cost relative to ASGS method is problem-dependent, the accuracy is usually higher, in particular in what concerns the pressure approximation near boundaries.