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

Abstract

This paper studies non inf-sup stable nite element approximations to the evolutionary Navier{Stokes equations. Several local projection stabilization (LPS) methods corresponding to di erent stabilization terms are analyzed, thereby separately studying the e ects of the di erent stabilization terms. Error estimates are derived in which the constants in the error bounds are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure nite elements of degree l, it will be proved that the velocity error in L1(0; T;L2( )) decays with rate l + 1=2 in the case that h, with being the dimensionless viscosity and h the mesh width. In the analysis of another method, it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies con rm the analytical results.