Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations

Abstract

In this paper, we analyze a class of globally divergence-free (and therefore pressure-robust) hybridizable discontinuous Galerkin (HDG) finite element methods for stationary Navier-Stokes equations. The methods use the $$\cal{P}_{k}/\cal{P}_{k-1}(k\geqslant 1)$$ discontinuous finite element combination for the velocity and pressure approximations in the interior of elements, piecewise $$\cal{P}_{m}(m=k,k-1)$$ for the velocity gradient approximation in the interior of elements, and piecewise $$\cal{P}_{k}/\cal{P}_{k}$$ for the trace approximations of the velocity and pressure on the inter-element boundaries. We show that the uniqueness condition for the discrete solution is guaranteed by that for the continuous solution together with a sufficiently small mesh size. Based on the derived discrete HDG Sobolev embedding properties, optimal error estimates are obtained. Numerical experiments are performed to verify the theoretical analysis.