Stabilised hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions
Provides a numerical method for solving Stokes problems with non-standard boundary conditions, but the contribution is incremental as it extends existing stabilisation techniques to a new discretisation framework.
The authors apply a stabilisation approach to hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions, achieving smaller pressure errors compared to inf-sup stable elements, though no formal proof is provided.
In several studies it has been observed that, when using stabilised $\mathbb{P}_k^{}\times\mathbb{P}_k^{}$ elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable $\mathbb{P}_k^{}\times\mathbb{P}_{k-1}^{}$ (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not stability the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions.