Mixed finite element methods: Implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods

In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart-Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart-Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.