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

Abstract

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 $L^{\infty }(I;L^2(\Omega ))$, $L^2(I;H^1(\Omega ))$ and $L^2(I;L^2(\Omega ))$ norms have been shown. The main result of the present work extends the error estimate in the $L^{\infty }(I;L^2(\Omega ))$ 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 $L^{\infty }(I;L^2(\Omega ))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\Omega ))$ and $L^2(I;L^2(\Omega ))$ norms, which complement this work.

revised version; added some detail to proofs; reordered some results for better readability