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.