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

Abstract

We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure interpolations. The method consists in introducing the project of the pressure gradient and adding the difference between the pressure Laplacian and divergence of this new field to the incompressibility equations, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new 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 or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for stationary and the transient of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP.