  We introduce a new concept of the locally conservative flux and investigate
its relationship with the compatible discretization pioneered by Dawson, Sun
and Wheeler [11]. We then demonstrate how the new concept of the locally
conservative flux can play a crucial role in obtaining the L2 norm stability of
the discontinuous Galerkin finite element scheme for the transport in the
coupled system with flow. In particular, the lowest order discontinuous
Galerkin finite element for the transport is shown to inherit the positivity
and maximum principle when the locally conservative flux is used, which has
been elusive for many years in literature. The theoretical results established
in this paper are based on the equivalence between Lesaint-Raviart
discontinuous Galerkin scheme and Brezzi-Marini-Suli discontinuous Galerkin
scheme for the linear hyperbolic system as well as the relationship between the
Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method
along the streamline. Sample numerical experiments have also been performed to
justify our theoretical findings
