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

翻译：本文引入了一种新的局部守恒通量概念，并探讨了其与Dawson、Sun和Wheeler [11] 开创的相容离散化之间的关系。随后，我们论证了该局部守恒通量概念如何在获得耦合流动-输运系统中输运方程间断有限元格式的L2范数稳定性方面发挥关键作用。特别地，当采用局部守恒通量时，输运方程的最低阶间断有限元格式被证明能够继承正性和最大原理，这一特性在文献中长期未能实现。本文建立的理论结果基于线性双曲系统中Lesaint-Raviart间断有限元格式与Brezzi-Marini-Suli间断有限元格式的等价性，以及Lesaint-Raviart间断有限元格式与沿流线特征法之间的关系。我们还进行了数值算例以验证理论结论。