Analysis of a family of continuous–discontinuous Galerkin FEM for convection–diffusion problems

It is well known that continuous Galerkin methods lack stability for singularly perturbed convection–diffusion problems. One approach to overcome this behaviour is to use discontinuous Galerkin methods instead. Unfortunately, this increases the number of degrees of freedom and thus the computational costs.We analyse discontinuous Galerkin methods of anisotropic polynomial order and discrete discontinuous spaces. By enforcing continuity in the vertices of a mesh, the number of unknowns can be reduced while the convergence order in the dG-norm is still sustained.Numerical experiments for several polynomial elements and finite element spaces support our theoretical results.