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This paper studies fully discrete finite element approximations to theNavier-Stokes equations using inf-sup stable elements and grad-divstabilization. For the time integration two implicit-explicit second orderbackward differentiation formulae (BDF2) schemes are applied. In both thelaplacian is implicit while the nonlinear term is explicit, in the first one,and semi-implicit, in the second one. The grad-div stabilization allow us toprove error bounds in which the constants are independent of inverse powers ofthe viscosity. Error bounds of order $r$ in space are obtained for the $L^2$error of the velocity using piecewise polynomials of degree $r$ to approximatethe velocity together with second order bounds in time, both for fixed timestep methods and for methods with variable time steps. A CFL-type condition isneeded for the method in which the nonlinear term is explicit relating timestep and spatial mesh sizes parameters.

Robust error bounds for the Navier-Stokes equations using implicit-explicit second order BDF method with variable steps