Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations

Abstract:

We present the first a priori error analysis of the hybridizable discontinuous Galerkin method for the approximation of the Navier-Stokes equations proposed in J. Comput. Phys. vol. 230 (2011), pp. 1147-1170. The method is defined on conforming meshes made of simplexes and provides piecewise polynomial approximations of fixed degree $k$ to each of the components of the velocity gradient, velocity and pressure. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global $L^2$-norm of the error in each of the above-mentioned variables converges with the optimal order of $k+1$ for $k\ge 0$. We also prove a superconvergence property of the velocity which allows us to obtain an elementwise postprocessed approximate velocity, $H(\textrm {div})$-conforming and divergence-free, which converges with order $k+2$ for $k\ge 1$. In addition, we show that these results only depend on the inverse of the stabilization parameter of the jump of the normal component of the velocity. Thus, if we superpenalize those jumps, these converegence results do hold by assuming that the pressure lies in $H^1(\Omega )$ only. Moreover, by letting such stabilization parameters go to infinity, we obtain new $H(\textrm {div})$-conforming methods with the above-mentioned convergence properties.