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Abstract We introduce a new concept of the local flux conservation and investigate its role in the coupled flow and transports. We demonstrate how the proposed concept of the locally conservative flux can play a crucial role in obtaining the$$L^2$$ $L^2$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-Süli 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 then been performed to justify our theoretical findings. 

Abstract We consider additive Schwarz methods for boundary value problems involving the $$p$$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. First, we present a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behaviour akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincaré–Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements we establish the asymptotic linear convergence of additive Schwarz methods for the $$p$$-Laplacian. 

Abstract We present an efficient numerical method to approximate the flux variable for the Darcy flow model. An important feature of our new method is that the approximate solution for the flux variable is obtained without approximating the pressure at all. To accomplish this, we introduce a user‐defined parameter delta, which is typically chosen to be small so that it minimizes the negative effect resulting from the absence of the pressure, such as inaccuracy in both the flux approximation and the mass conservation. The resulting algebraic system is of significantly smaller degrees of freedom, compared to the one from the mixed finite element methods or least‐squares methods. We also interpret the proposed method as a single step iterate of the augmented Lagrangian Uzawa applied to solve the mixed finite element in a special setting. Lastly, the pressure recovery from the flux variable is discussed and an optimal‐order error estimate for the method is obtained. Several examples are provided to verify the proposed theory and algorithm, some of which are from more realistic models such as SPE10.