Rellich-type Discrete Compactness for Some Discontinuous Galerkin FEM

We deduce discrete compactness of Rellich type for some discontinuous Galerkin finite element methods (DGFEM) including hybrid ones, under fairly general settings on the triangulations and the finite element spaces. We make use of regularity of the solutions to an auxiliary second-order elliptic boundary value problem as well as the error estimates of the associated finite element solutions. The present results can be used for analyzing DGFEM applied to some boundary value and eigenvalue problems, and also to derive the discrete Poincar´e-Friedrichs inequalities.