an encyclopedia of finite element definitions

# Serendipity H(curl)

 Alternative names Brezzi–Douglas–Marini cubical H(curl) (quadrilateral), Arnold–Awanou H(curl) (hexahedron) De Rham complex families $$\left[S_{1,k}^\square\right]_{1}$$ / $$\mathcal{S}_{k}\Lambda^{1}(\square_d)$$ Abbreviated names BDMce (quadrilateral), AAe (hexahedron) Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(27)}_{k}$$ (quadrilateral) $$\mathcal{P}_{k}^d \oplus \mathcal{A}_{k-1} \oplus \mathcal{Z}^{(28)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each edge: tangent integral moments with an order $$k$$ dPc space On each face: integral moments with an order $$k-2$$ vector dPc space On each volume: integral moments with an order $$k-4$$ vector dPc space Number of DOFs quadrilateral: $$k^2+3k+4$$ (A014206)hexahedron: $$\begin{cases}6(k^2+k+2)&k=1,2,3\\k(k+1)(k-1)/2 + 3k^2 + 12k + 9&k > 3\end{cases}$$ Mapping covariant Piola continuity Components tangential to facets are continuous Categories Vector-valued elements, H(curl) conforming elements

## Implementations

 Symfem "Scurl"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "BDMCE" (quadrilateral)"AAE" (hexahedron)↓ Show (legacy) UFL examples ↓

## Examples

 quadrilateralorder 1 (click to view basis functions) quadrilateralorder 2 (click to view basis functions) hexahedronorder 1 (click to view basis functions) hexahedronorder 2 (click to view basis functions)

## DefElement stats

 Element added 31 December 2020 Element last updated 16 September 2023