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

Chair of Optimal Control, Technical University of Munich, School of Computation Information and Technology, Department of Mathematics, Boltzmannstraße 3, 85748 Garching bei Munich, Germany

^* Corresponding author: wagnerja@cit.tum.de

Received: 26 June 2023
Accepted: 23 January 2024

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^∞(I; L²(Ω)), L²(I; H¹(Ω)) and L²(I; L²(Ω)) norms have been shown. The main result of the present work extends the error estimate in the L^∞(I; L²(Ω)) 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^∞(I; L²(Ω)) error estimate, also allow us to show best approximation type error estimates in the L²(I; H¹(Ω)) and L²(I; L²(Ω)) norms, which complement this work.