Mass-lumped mixed finite elements methods and the abstract gradient scheme framework for elliptic problems

We provide a general framework for second-order elliptic problems, which includes a variety of boundary conditions. We show how one can apply mass-lumped mixed finite element to this problem, and we provide sufficient conditions for the convergence of such a method. In particular, we exhibit convergence results assuming two different type of assumptions: on one hand, we show convergence properties following the standard analysis of mixed finite elements. On the other hand, we provide conditions on one of the approximation space, which also lead to some convergence properties. We then formulate Abstract Gradient Discretization Method (AGDM) based on these mass-lumped mixed finite elements, enabling us to apply this type of discretization to a variety of nonlinear problems. Finally, we illustrate our results by three examples. The first one is a second-order elliptic problem with homogeneous Neumann boundary conditions, discretized by Raviart-Thomas finite elements. We show on this example that mass-lumping leads to classical finite volume schemes. The second one concerns the same problem but with a more general elliptic operator. The third one is inspired by the elliptic part of a model of shallow water flows with dispersive terms. We apply on this example generalized operators, and we prove the convergence of the method used in the literature.