Local Projection Method for Inf-Sup Stable Elements

The previous chapter showed that local projection stabilization (LPS) for equal-order interpolation can handle two types of instabilities — that caused by a violation of the discrete inf-sup condition and that due to dominant convection in the case of high Reynolds number. But the flow problem is often only part of a coupled flow-transport problem; in the next chapter we shall see that mass conservation in the transport equation depends on the properties of the discrete velocity and in particular on the satisfaction of the incompressibility constraint. Unfortunately, when LPS is applied with equalorder interpolation, the discrete divergence-free property of the velocity field is disturbed by the term

$$\sum \limits_{M\in{\cal M}_h} \alpha_M(\kappa_h\nabla{p_h}, \kappa_h \nabla{q_h}) M$$

that stabilizes the pressure. For inf-sup stable finite element pairs, this pressure stabilization is unnecessary and we are faced only with the instability caused by dominant convection. Thus it is of interest to consider local projection stabilization for inf-sup stable finite elements.

The main objective of this chapter is an analysis of convergence properties of LPS applied to inf-sup stable discretizations of the Oseen problem.We shall restrict our attention to the enrichment variant of LPS and to a stabilizing term that controls separately fluctuations of the derivative in the streamline direction and fluctuations of the divergence.