  We study the convergence of the discontinuous Galerkin (DG) method applied to
the advection-reaction equation on meshes with reentrant faces. On such meshes,
the upwind numerical flux is not smooth, and so the numerical integration of
the resulting face terms can only be expected to be first-order accurate.
Despite this inexact integration, we prove that the DG method converges with
order $\mathcal{O}(h^{p+1/2})$, which is the same rate as in the case of exact
integration. Consequently, specialized quadrature rules that accurately
integrate the non-smooth numerical fluxes are not required for high-order
accuracy. These results are numerically corroborated on examples of linear
advection and discrete ordinates transport equations.
