Error estimates for finite element discretizations of the instationary Navier-Stokes equations

In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the L infinity(I; L2(omega)), L2(I; H1(omega)) and L2(I; L2(omega)) norms have been shown. The main result of the present work extends the error estimate in the L infinity(I; L2(omega)) norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the L infinity(I; L2(omega)) error estimate, also allow us to show best approximation type error estimates in the L2(I; H1(omega)) and L2(I; L2(omega)) norms, which complement this work.