2023/07/26 by Vexler, Boris, Wagner, Jakob · 1 citation
#35Q30 #65M15 #65M22 #65M60 #76D05 #76M10 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2307.14217
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.