Two Families of $H$(div) Mixed Finite Elements on Quadrilaterals of Minimal Dimension Article Swipe
YOU?
·
· 2016
· Open Access
·
· DOI: https://doi.org/10.1137/15m1013705
· OA: W2556087308
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 (${\mathbf u}$, $p$, and $\nabla\cdot{\mathbf u}$ are approximated to the same order like RT) and the other has reduced $H(div)$-approximation ($p$ and $\nabla\cdot{\mathbf u}$ 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 $\nabla\cdot{\mathbf u}$ 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.
