  In this study, we consider the simulation of subsurface flow and solute
transport processes in the stationary limit. In the convection-dominant case,
the numerical solution of the transport problem may exhibit non-physical
diffusion and under- and overshoots. For an interior penalty discontinuous
Galerkin (DG) discretization, we present a $h$-adaptive refinement strategy
and, alternatively, a new efficient approach for reducing numerical under- and
overshoots using a diffusive $L^2$-projection. Furthermore, we illustrate an
efficient way of solving the linear system arising from the DG discretization.
In $2$-D and $3$-D examples, we compare the DG-based methods to the streamline
diffusion approach with respect to computing time and their ability to resolve
steep fronts.
