The authors consider discretizations of the Poisson equation subject to homogeneous Dirichlet boundary conditions in a bounded open connected polytopal set in \(\mathbb{R}^d\), \(d \geq 1\). Two families of discretizations are studied: the mixed and the primal formulation of discontinuous skeletal methods. The denotation “skeletal” comes from the feature that the set of degrees of freedom (short hand: DOF’s) corresponding to inside mesh elements is eliminated by static condensation and, as a consequence, only those on the mesh skeleton (which are responsible for the transmission of information) are left in the equations. The polynomials in skeletal methods are continuous in the interface points but discontinuous at vertices or edges, respectively.
Several methods existing in the literature are shown to belong to one of the skeletal discretizations. One aim of the paper is a unified formulation of the two skeletal discretizations. It is shown how to obtain from a given skeletal method in mixed formulation an equivalent method in primal formulation. The converse can be shown under an assumption on the kind of approximation of the gradients. A convergence analysis is given for the unified formulation providing optimal \(L^2\)- and energy-norm error bounds.