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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. Schoberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an $H({div})$-conforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. $H({div})$-conformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree $k$, it requires unknowns of degree $k$ on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree $k-1$ on the facets are required to obtain an accurate polynomial approximation of order $k$ (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the $H({div})$-conformity so that only unknowns of polynomial degree $k-1$ are involved for normal co...

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