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Serendipity H(curl)

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Alternative namesBrezzi–Douglas–Marini cubical H(curl) (quadrilateral), Arnold–Awanou H(curl) (hexahedron)
Exterior calculus names\(\mathcal{S}_{k}\Lambda^{1}(\square_d)\)
Cockburn–fu names\(\left[S_{1,k}^\square\right]_{1}\)
Abbreviated namesBDMce (quadrilateral), AAe (hexahedron)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(36)}_{k}\) (quadrilateral)
\(\mathcal{P}_{k}^d \oplus \mathcal{A}_{k-1} \oplus \mathcal{Z}^{(37)}_{k}\) (hexahedron)
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DOFsOn 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 DOFsquadrilateral: \(k^2+3k+4\) (A014206)
hexahedron: \(\begin{cases}6(k^2+k+2)&k=1,2,3\\k(k+1)(k-1)/2 + 3k^2 + 12k + 9&k > 3\end{cases}\)
Mappingcovariant Piola
continuityComponents tandential to facets are continuous
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

Symfem"Scurl"
↓ Show Symfem examples ↓
UFL"BDMCE" (quadrilateral)
"AAE" (hexahedron)
↓ Show UFL examples ↓

Examples

quadrilateral
order 1

(click to view basis functions)
quadrilateral
order 2

(click to view basis functions)
hexahedron
order 1

(click to view basis functions)
hexahedron
order 2

(click to view basis functions)

References

DefElement stats

Element added31 December 2020
Element last updated02 August 2022