Local Projection Stabilization for Equal-Order Interpolation

Local projection stabilization (LPS) was introduced in Part III, Chapter 3 for a scalar convection-diffusion equation. It will now be extended to the Oseen system

$$-\nu \Delta {\bf u} + ({\bf b} \cdot \nabla){\bf u} + \sigma {\bf u} + \nabla p = {\bf f} \; {\rm in} \; \Omega \subset {{\bf R}^d}$$

((4.1a))

$$\nabla \cdot {\bf u} = 0 \; {\rm in} \; \Omega$$

((4.1b))

$${\bf u} = {\bf 0} \; {\rm on} \; \partial\Omega$$

((4.1c))

As we saw in Chapter 3, the streamline diffusion method (SDFEM) can handle two types of instabilities: that caused by a violation of the discrete infsup (Babu?ska-Brezzi) condition (2.1) and that due to dominant convection. The SDFEM combines the Pressure Stabilized Petrov-Galerkin (PSPG) approach (testing the residual against ${\nabla{\rm q}}$) with the Streamline Upwind Petrov- Galerkin (SUPG) technique (testing the residual against ${({\rm b}\cdot\nabla)}{\bf v}$)

Despite the extensive theoretical and practical development of the SDFEM, a fundamental flaw in the method - in particular for higher-order interpolations - is that various terms must be added to the weak formulation to guarantee its consistency. Moreover, the requirement of consistency leads to undesirable effects when using residual-based stabilization methods like the SDFEM in optimal control problems; see the discussion at the beginning of Section III.3.3. LPS relaxes the consistency requirement while preserving the main features of the SDFEM approach; in particular, one can use equal-order interpolation without worrying about the Babuška-Brezzi condition. Furthermore, LPS allows us to separate velocity and pressure in the stabilization terms, which for systems of equations means that one can avoid non-physical couplings.