A finite element penalty–projection method for incompressible flows

The penalty–projection method for the solution of Navier–Stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage, namely the solution of the momentum balance equation, to constrain the divergence of the predicted velocity field. After a presentation of the scheme in the time semi-discrete formulation, then in fully discrete form for a finite element discretization, we assess its behaviour against a set of benchmark tests, including in particular prescribed velocity and open boundary conditions. The results demonstrate that the augmentation always produces beneficial effects. As soon as the augmentation parameter takes a significant value, the projection method splitting error is reduced, pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore. For high values of the augmentation parameter, the results of coupled solvers are recovered. Consequently, in contrast with standard penalty methods, there is no need for a dependence of the augmentation parameter with the time step, and this latter can be kept to reasonable values, to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step.