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This paper studies non inf-sup stable finite element approximations to theevolutionary Navier--Stokes equations. Several local projection stabilization(LPS) methods corresponding to different stabilization terms are analyzed,thereby separately studying the effects of the different stabilization terms.Error estimates are derived in which the constants in the error bounds areindependent of inverse powers of the viscosity. For one of the methods, usingvelocity and pressure finite elements of degree $l$, it will be proved that thevelocity error in $L^\infty(0,T;L^2(\Omega))$ decays with rate $l+1/2$ in thecase that $\nu\le h$, with $\nu$ being the dimensionless viscosity and $h$ themesh width. In the analysis of another method, it was observed that theconvective term can be bounded in an optimal way with the LPS stabilization ofthe pressure gradient. Numerical studies confirm the analytical results.

Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier--Stokes Equations with Local Projection Stabilization