Some remarks on quadrilateral mixed finite elements

It is well known that quadrilateral finite elements may produce unsatisfactory results when used on distorted meshes. It turns out that many commonly used finite elements achieve suboptimal convergence properties on distorted quadrilaterals; among such elements we recall in particular serendipity (trunk) scalar elements and basically all vectorial elements for the approximation of problems involving the functional space image (like Raviart–Thomas or Brezzi–Douglas–Marini spaces for Darcy flow). In two space dimensions, a similar remark applies to edge finite elements for the approximation of Maxwell’s problems involving the space image. On the other hand, mimetic finite differences have become popular for the approximation of problems involving image on very general geometries. The aim of this paper is to show how to use the ideas of mimetic finite differences for the stabilization of Raviart–Thomas element on general quadrilateral meshes. It turns out that such stabilization can be performed by a slight modification of the standard Raviart–Thomas element which does not increase significantly the computational cost of the original scheme.