Two Families of H(div) Mixed Finite Elements on Quadrilaterals of Minimal Dimension

We develop two families of mixed finite elements on quadrilateral meshes for approximating ( u,p) solving a second order elliptic equation in mixed form. Standard Raviart--Thomas (RT) and Brezzi--Douglas--Marini (BDM) elements are defined on rectangles and extended to quadrilaterals using the Piola transform, which are well-known to lose optimal approximation of ∇· u. Arnold--Boffi--Falk spaces rectify the problem by increasing the dimension of RT, so that approximation is maintained after Piola mapping. Our two families of finite elements are uniformly inf-sup stable, achieve optimal rates of convergence, and have minimal dimension. The elements for u are constructed from vector polynomials defined directly on the quadrilaterals, rather than being transformed from a reference rectangle by the Piola mapping, and then supplemented by two (one for the lowest order) basis functions that are Piola mapped. One family has full $H(div)$-approximation (u, p, and ∇· u are approximated to the same order like RT) and the other has reduced $H(div)$-approximation (p and ∇· u are approximated to one less power like BDM). The two families are identical except for inclusion of a minimal set of vector and scalar polynomials needed for higher order approximation of ∇· u and p, and thereby we clarify and unify the treatment of finite element approximation between these two classes. The key result is a Helmholtz-like decomposition of vector polynomials, which explains precisely how a divergence is approximated locally. We develop an implementable local basis and present numerical results confirming the theory.