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

Martin Vohralík (1) , Barbara Wohlmuth (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 LNM - Lehrstuhl fuer Numerische Mathematik

In this paper, we study the mixed finite element method. We focus on the lowest-order Raviart--Thomas case and propose several new ways of reducing the original indefinite saddle-point systems for flux and potential unknowns to (positive definite) systems for potential unknowns only. It turns out that the principle of our construction is closely related to that of the so-called multi-point flux-approximation method. We study theoretically these different ways and illustrate them by a series of numerical test examples. We also discuss the different versions of the discrete maximum principle and of obtaining local flux expressions in the mixed finite element method. We finally recall the use of the mixed finite element method on general polygonal meshes and expose its similarity to mimetic finite difference, multi-point flux-approximation, mixed finite volume, and hybrid finite volume methods in this case.

Mots clés: Mixed finite element method; Reformulation; One unknown per element; Local flux expression; Discrete maximum principle; Polygonal mesh; Relations between different methods

Domaines: Analyse numérique [math.NA]

Martin Vohralík, Barbara Wohlmuth. Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. 2010. ⟨hal-00497394v2⟩