Semirobust analysis of an H(div)-conforming DG method with semi-implicit time-marching for the evolutionary incompressible Navier–Stokes equations

Abstract

Abstract In this paper, we present a fully discrete analysis of an H(div)-conforming DG method with semi-implicit time-marching for the evolutionary incompressible Navier–Stokes equations. We use a semi-implicit time-discrete scheme in which the convection velocity is treated explicitly for the convection term. A stability analysis and a priori error estimates are given, in which the constants are independent of the negative powers of the viscosity. For inf-sup stable H(div)-conforming finite element pairs $BDM_k/P_{k-1}$ and $RT_k/P_k$, the rate of convergence $k+1/2$ is proved for the $L^2$ error of the velocity in the case of $\nu \leq C h$, where $k$ is the degree of the polynomials in the velocity approximation. In particular, for the inf-sup stable finite element pair $RT_k/P_k$, the convergence rate of the pressure is also $k+1/2$ when $\nu \leq C h$. The numerical experiments verify the analytical results.