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 \mathbf 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 \nabla\cdot{\mathbf 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 \mathbf 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 \mathbf u} $p$
SIAM J. Numer. Anal.