Gradient-robust hybrid DG discretizations for the compressible Stokes equations (Q6594651)

As a model problem the compressible Stokes equations \(- \nu \Delta u + \nabla (c_M \varrho) = \varrho g +f\), \(\mathrm{div}(\varrho u) = 0\) are considered, where \(u\) is the velocity with homogeneous Dirichlet boundary data, \(g\) a given gravity force, \(f\) an additional force, \(\nu\) the viscosity, \(\varrho\) the density, and \(c_M\) a positive constant. A weak formulation of this problem is given. A hybrid discontinuous Galerkin scheme is proposed, where the space for the discrete velocity is the \(H(\mathrm{div})\)-conforming Brezzi-Douglas-Marini-space of order \(k\). For the density, a discontinuous ansatz space is used. The gradient-robustness of this scheme, stability, and the existence of solutions are proved. Additionally, positivity and mass preserving of the discrete density are shown in the case of piecewise constant approximations. Convergence of the discrete solutions is proved. Furthermore, a full hybrid discontinuous Galerkin scheme is presented. For this scheme also stability and convergence are proved. The two presented schemes are compared with respect to convergence by means of three numerical examples.