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 ≥ 0. We also prove a superconvergence property of the velocity which allows us to obtain an elementwise postprocessed approximate velocity, H(div)-conforming and divergence-free, which converges with order k + 2 for k ≥ 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 (Ω) only. Moreover, by letting such stabilization parameters go to infinity, we obtain new H(div)-conforming methods with the above-mentioned convergence properties.

A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

“…This method uses P k (T h ), row-wise k-th order RT space and P k (T h ) to approximate the velocity gradient, stress and velocity, respectively. The convergence of the velocity gradient and velocity in L 2 -norm and the stress in H(div)-norm is of order k. More recently, in 2015, Cockburn et al [6] gave an error analysis of the HDG method developed in [23] which is close to method in this paper. Our method may be criticized by the fact that with the modification of the scheme, we no longer have the local conservation of the momentum.…”

“…Our formulation is close to that of the HDG method in [6,23], in which they have the same global degrees of freedom u h . Nevertheless, there are three crucial differences which lead to special properties of our HDG method.…”

“…To define the HDG method, we adopt the notations and norms used in [6]. We consider conforming triangulation T h of Ω made of shape-regular polyhedral elements which can be non-convex.…”

“…Firstly, our method uses P k+1 (T h ) to approximate the primary variable, which is the velocity, on each element while methods in [6,23] uses P k (T h ) instead. Secondly, the diffusion part of the stabilization function ν h (Π M u h − u h ) in (1.2e) is totally different from those used in [6,23]. Finally, motivated by the work in [29] and [8], we insert two terms −(…”

“…Our method may be criticized by the fact that with the modification of the scheme, we no longer have the local conservation of the momentum. Nevertheless, our approach has several advantages comparing with the one in [6,23]. For instance, the analysis in [6,23] is only valid for simplicical meshes and it needs a postprocessing procedure to obtain superconvergent approximation to the velocity.…”