  In this paper, we propose and analyze a numerically stable and convergent
scheme for a convection-diffusion-reaction equation in the convection-dominated
regime. Discontinuous Galerkin (DG) methods are considered since standard
finite element methods for the convection-dominated equation cause spurious
oscillations. We choose to follow a novel DG finite element differential
calculus framework introduced in Feng et al. (2016) and approximate the
infinite-dimensional operators in the equation with the finite-dimensional DG
differential operators. Specifically, we construct the numerical method by
using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive
term and the average discrete gradient operator for the convective term along
with standard DG stabilization. We prove that the method converges optimally in
the convection-dominated regime. Numerical results are provided to support the
theoretical findings.
