Finite Elements with Local Projection Stabilization for Incompressible Flow Problems (Journal of Computational Mathematics, 2018)
In this paper we review recent developments in the analysis of finite element methodsfor incompressible flow problems with local projection stabilization (LPS). These methodspreserve the favourable stability and approximation properties of classical residual-basedstabilization (RBS) techniques but avoid the strong coupling of velocity and pressure in thestabilization terms. LPS-methods belong to the class of symmetric stabilization techniquesand may be characterized as variational multiscale methods. In this work we summarizethe most important a priori estimates of this class of stabilization schemes developed in thepast 6 years. We consider the Stokes equations, the Oseen linearization and the Navier-Stokes equations. Furthermore, we apply it to optimal control problems with linear(ized)flow problems, since the symmetry of the stabilization leads to the nice feature that theoperations "discretize" and "optimize" commute.