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

TL;DR

This paper derives error estimates for fully discrete finite element methods applied to the 2D instationary Navier-Stokes equations, extending previous results to include the $L^ abla(I;L^2( abla))$ norm.

Contribution

It extends best approximation error estimates to the Navier-Stokes equations using a duality approach and a discrete Gronwall lemma.

Findings

Error estimates in the $L^ abla(I;L^2( abla))$ norm for Navier-Stokes

Error splitting and duality techniques enable new bounds

Stability analysis via a tailored discrete Gronwall lemma

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.