# Pressure Stabilization#

```Pressure Stabilization = {yes | no | local | pspp | pspp_e}
```

## Description / Usage#

This optional card indicates whether or not pressure stabilization should be used. Valid options are

yes

Use the Galerkin Least square pressure stabilization method developed by Hughes, et. al. (1986).

local

Use the Galerkin Least square pressure stabilization method with local scaling.

pspp

Use polynomial stabilized pressure projection stabilization method developed by Dohrmann and Bochev (2004). Please see Level Set PSPP filtering card if using with the level-set front tracking technique.

pspp_e

Use polynomial stabilized pressure projection method with upgrade for nonuniform/graded meshes (recommended)

no

Do not use any pressure stabilization.

The amount of pressure stabilization to use is specified with the Pressure Stabilization Scaling card.

The default is no, to not use pressure stabilization.

## Examples#

Following is a sample card:

```Pressure Stabilization = yes
```

## Technical Discussion#

If input for this card is yes, the Hughes, et al. (1986) method adds the residual of the momentum equation weighted by the gradient of the Galerkin weight function to the Galerkin continuity equation. The result is that the continuity equation now has a diagonal term to stabilize it and improve the condition of the matrix, allowing for the use of iterative solvers. When pressure stabilization is used, equal-order interpolation can (and should) be used for velocity and pressure, e.g., velocity and pressure both Q2 or both Q1. If input for this card is no, then the standard Galerkin finite-element weight functions are used and velocity and pressure interpolations should be chosen to satisfy the Babuska-Brezzi condition, e.g., velocity Q2 and pressure Q1 or P1, or velocity Q1 and pressure P0.

An improvement on the Hughes approach was developed by Bochev and Dohrmann (2004) called the polynomial stabilized pressure projection. In its fundamental form, it is like PSPG just an additional term on the continuity equation residual that helps stabilize the pressure, and it is predicated on the fact that the pressure field is governed by an elliptical equation known as the pressure Poisson equation. Please consult this paper for details. An additional improvement to that technique was developed internally to Sandia which better accommodates graded meshes. This technique is invoked with the pspp_e option, which we recommend.

## References#

Hughes, T. J. R., L. P. Franca and M. Balestra, “A New Finite Element Formulation for Computational Fluid Dynamics: V. Circumventing the Babuska-Brezzi Condition: A Stable Petrov-Galerkin Formulation of the Stokes Problem Accommodating Equal- Order Interpolations,” Comput. Methods Appl. Mech. Engrg., 59 (1986) 85-99.