Error analysis of BDF schemes for the evolutionary incompressible Navier--Stokes equations

TL;DR

This paper derives error bounds for fully discrete BDF schemes (up to order 5) applied to the evolutionary incompressible Navier--Stokes equations, including standard and grad-div stabilized methods, with results on convergence rates.

Contribution

It provides the first error bounds for BDF-$q$ schemes with $q extgreater= 3$ for Navier--Stokes equations, including analysis of grad-div stabilization.

Findings

Optimal error bounds for velocity and pressure with order $( riangle t)^q$ in time.

Error bounds with spatial convergence rates $h^{k+1}$ and $h^k$ for velocity in $L^2( ext{domain})$.

Grad-div stabilization yields bounds independent of inverse viscosity.

Abstract

Error bounds for fully discrete schemes for the evolutionary incompressible Navier--Stokes equations are derived in this paper. For the time integration we apply BDF-$q$ methods, $q\le 5$, for which error bounds for $q\ge 3$ cannot be found in the literature. Inf-sup stable mixed finite elements are used as spatial approximation. First, we analyze the standard Galerkin method and second a grad-div stabilized method. The grad-div stabilization allows to prove error bounds with constants independent of inverse powers of the viscosity coefficient. We prove optimal bounds for the velocity and pressure with order $(\Delta t)^q$ in time for the BDF-$q$ scheme and order $h^{k+1}$ for the $L^2(\Omega)$ error of the velocity in the first case and $h^k$ in the second case, $k$ being the degree of the polynomials in finite element velocity space.