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Serendipity H(div)

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Alternative namesBrezzi–Douglas–Marini cubical H(div) (quadrilateral), Arnold–Awanou H(div) (hexahedron)
Exterior calculus names\(\mathcal{S}_{k}\Lambda^{d-1}(\square_d)\)
Cockburn–fu names\(\left[S_{1,k}^\square\right]_{d-1}\)
Abbreviated namesBDMcf (quadrilateral), AAf (hexahedron)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(17)}_{k}\) (quadrilateral)
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(18)}_{k}\) (hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an order \(k\) dPc space
On the interior of the reference element: integral moments with an order \(k-2\) vector dPc space
Number of DOFsquadrilateral: \(k^2+3k+4\) (A014206)
hexahedron: \((k+1)(k^2+5k+12)/2\)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"Sdiv"
↓ Show Symfem examples ↓
UFL"BDMCF" (quadrilateral)
"AAF" (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 added30 December 2020
Element last updated02 August 2022