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Trimmed serendipity H(div)

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De Rham complex families\(\left[S_{2,k}^\square\right]_{d-1}\) or \(\mathcal{S}^-_{k}\Lambda^{d-1}(\square_d)\)
Degrees\(0\leqslant k\)
where \(k\) is the Polynomial subdegree
Polynomial subdegree\(k\)
Polynomial superdegree\(k+1\)
Lagrange subdegree\(\operatorname{floor}((k+2)/(d+1))\)
Lagrange superdegree\(k+1\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(47)}_{k+1} \oplus \mathcal{Z}^{(48)}_{k+1}\) (quadrilateral)
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(47)}_{k+1} \oplus \mathcal{Z}^{(49)}_{k+1} \oplus \mathcal{Z}^{(50)}_{k+1} \oplus \mathcal{Z}^{(51)}_{k+1}\) (hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with a degree \(k\) dPc space
On the interior of the reference element: integral moments with a degree \(k-2\) vector dPc space, and integral moments with \(\left\{\nabla(p)\middle|p\text{ is a degree \(k\) monomial}\right\}\)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

This element is implemented in FIAT and Symfem .↓ Show implementation detail ↓

Examples

quadrilateral
degree 0

(click to view basis functions)
quadrilateral
degree 1

(click to view basis functions)
quadrilateral
degree 2

(click to view basis functions)
hexahedron
degree 0

(click to view basis functions)
hexahedron
degree 1

(click to view basis functions)
hexahedron
degree 2

(click to view basis functions)

References

DefElement stats

Element added07 October 2021
Element last updated31 March 2025