Error analysis of BDF schemes for the evolutionary incompressible Navier--Stokes equations

Error bounds for fully discrete schemes for the evolutionary incompressible Navier–Stokes equations are derived in this paper. For the time integration we apply BDF-$q$ methods, $q\le 5$, for which error bounds for $q\ge 3$ cannot be found in the literature. Inf-sup stable mixed finite elements are used as spatial approximation. First, we analyze the standard Galerkin method and second a grad-div stabilized method. The grad-div stabilization allows to prove error bounds with constants independent of inverse powers of the viscosity coefficient. We prove optimal bounds for the velocity and pressure with order $(\mathrm{\Delta }t{)}^q$ in time for the BDF-$q$ scheme and order $h^{k+1}$ for the $L^2(\mathrm{\Omega })$ error of the velocity in the first case and $h^k$ in the second case, $k$ being the degree of the polynomials in finite element velocity space.