Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport

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