Error estimates for finite element discretizations of the instationary Navier-Stokes equations
Provides rigorous error analysis for numerical solutions of the Navier-Stokes equations, benefiting computational fluid dynamics researchers.
This work extends best approximation type error estimates for the Stokes problem to the instationary Navier-Stokes equations, proving error estimates in the L∞(I;L²(Ω)), L²(I;H¹(Ω)), and L²(I;L²(Ω)) norms for a fully discrete scheme using discontinuous Galerkin in time and inf-sup stable finite elements in space.
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 $L^\infty(I;L^2(Ω))$, $L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(Ω))$ 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 $L^\infty(I;L^2(Ω))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ norms, which complement this work.