The CG1–DG2 method for convection–diffusion equations in 2D

In this paper, we present the CG1–DG2 method for convection–diffusion equations. The space of continuous piecewise-linear functions is enriched with discontinuous quadratics so that the resultant finite element approximation is continuous at the vertices of the mesh but may have jumps across the edges. Three different approaches to the discretization of the diffusive part are considered: the symmetric interior penalty Galerkin method, the non-symmetric interior penalty Galerkin method and the Baumann–Oden method. In the context of elliptic problems we summarize well-known a priori error estimates for the discontinuous Galerkin approximation which carry over to the CG1–DG2 approach. Both methods have the same convergence rate which is also confirmed by numerical studies for diffusion and convection–diffusion problems.