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 piece- wise 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 L2-norm of the error in each of the We also prove a superconvergence property of the velocity which allows us to above-mentioned variables converges with the optimal order of k+1 for k ≥ 0. obtain an elementwise postprocessed approximate velocity, H(div)-conforming we show that these results only depend on the inverse of the stabilization pa- and divergence-free, which converges with order k +2 for k ≥ 1. In addition, rameter 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 H1(Ω) only. Moreover, by letting such stabilization parameters go to infinity, we obtain new H(div)-conforming methods with the above-mentioned convergence properties. 1.