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Serendipity

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Exterior calculus names\(\mathcal{S}_{k}\Lambda^{0}(\square_d)\), \(\mathcal{S}^-_{k-d}\Lambda^{0}(\square_d)\)
Cockburn–fu names\(\left[S_{1,k}^\square\right]_{0}\), \(\left[S_{2,k-d}^\square\right]_{0}\)
Abbreviated namesS
Orders\(1\leqslant k\)
Reference elementsinterval, quadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k} \oplus \mathcal{X}_{k}\)
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DOFsOn each vertex: point evaluations
On each edge: integral moments with an order \(k-2\) dPc space
On each face: integral moments with an order \(k-4\) dPc space
On each volume: integral moments with an order \(k-6\) dPc space
Number of DOFsinterval: \(k+1\) (A000027)
quadrilateral: \(\begin{cases}4&k=1\\k(k+3)/2+3&k>1\end{cases}\) (A340266)
hexahedron: \(\begin{cases}12k-4&k=1,2,3\\3k^2-3k+14&k=4,5\\k(k-1)(k+1)/6+k^2+5k+4&k>6\end{cases}\)
Mappingidentity
continuityFunction values are continuous.
CategoriesScalar-valued elements

Implementations

Basixbasix.ElementFamily.serendipity, ..., basix.LagrangeVariant.equispaced
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Symfem"serendipity"
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UFL"S"
↓ Show UFL examples ↓

Examples

interval
order 1

(click to view basis functions)
interval
order 2

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interval
order 3

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quadrilateral
order 1

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quadrilateral
order 2

(click to view basis functions)
quadrilateral
order 3

(click to view basis functions)

References

DefElement stats

Element added02 January 2021
Element last updated02 August 2022