Hybrid Discontinuous Galerkin Methods with Relaxed H(div)-Conformity for Incompressible Flows. Part I

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld and J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an ({div})ehBconforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. ({div})ehBconformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree $, it requires unknowns of degree $ on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree -1$ on the facets are required to obtain an accurate polynomial approximation of order $ (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the ({div})ehBconformity so that only unknowns of polynomial degree -1$ are involved for normal continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of ({div})ehBconformity, we introduce a cheap reconstruction operator which restores pressure robustness and pointwise divergence-free solutions and suits well to the finite element space with relaxed ({div})ehBconformity. We present this new method, carry out a thorough ehBversion error analysis, and demonstrate the performance of the method on numerical examples. Read More: https://epubs.siam.org/doi/10.1137/17M1138078