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 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 continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of H(div)-conformity, 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 H(div)-conformity. We present this new method, carry out a thorough h-version error analysis, and demonstrate the performance of the method on numerical examples.