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

Abstract This paper studies fully discrete finite element approximations to the Navier–Stokes equations using inf-sup stable elements and grad-div stabilization. For time integration, two implicit–explicit second-order backward differentiation formulae (BDF2) schemes are applied. In both, Laplacian is implicit while nonlinear term explicit, in first one, semiimplicit, second one. The stabilization allows us prove error bounds which constants independent of inverse powers viscosity. Error order $r$ space obtained for $L^2$ velocity piecewise polynomials degree approximate together with time, both fixed time-step methods variable steps. A Courant Friedrichs Lewy (CFL)-type condition needed method explicit relating spatial mesh-size parameters.