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Trimmed serendipity H(div)

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Exterior calculus names\(\mathcal{S}^-_{k}\Lambda^{d-1}(\square_d)\)
Cockburn–fu names\(\left[S_{2,k}^\square\right]_{d-1}\)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(5)}_{k} \oplus \mathcal{Z}^{(6)}_{k}\) (quadrilateral)
\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(5)}_{k} \oplus \mathcal{Z}^{(7)}_{k} \oplus \mathcal{Z}^{(8)}_{k} \oplus \mathcal{Z}^{(9)}_{k}\) (hexahedron)
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DOFsOn each facet: normal integral moments with an order \(k-1\) dPc space
On the interior of the reference element: integral moments with an order \(k-3\) vector dPc space, and integral moments with \(\left\{\nabla(p)\middle|p\text{ is an order \(k-1\) monomial}\right\}\)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"TSdiv"
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Examples

quadrilateral
order 1

(click to view basis functions)
quadrilateral
order 2

(click to view basis functions)
quadrilateral
order 3

(click to view basis functions)
hexahedron
order 1

(click to view basis functions)
hexahedron
order 2

(click to view basis functions)
hexahedron
order 3

(click to view basis functions)

References

DefElement stats

Element added07 October 2021
Element last updated02 August 2022