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

We discuss in this paper some implementation aspects of a finite element fo rmulation for the incompressible Navier-Stokes equations which allows the u se of equal order velocity-pressure interpolations. The method consists in introducing the projection of the pressure gradient and adding the differen ce between the pressure Laplacian and the divergence of this new field to t he incompressibility equation, both multiplied by suitable algorithmic para meters. The main purpose of this paper is to discuss how to deal with the n ew variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way for the algorithm to be eff icient is to uncouple it from the velocity-pressure calculation in one way or another. Here we discuss some iterative schemes to perform this uncoupli ng of the pressure gradient projection (PGP) from the calculation of the ve locity and the pressure, both for the stationary and the transient Navier-S tokes equations. In the first case, the strategies analyzed refer to the in teraction of the linearization loop and the iterative segregation of the PG P, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP. Copyright (C) 2001 John Wiley & Sons, Ltd.