an encyclopedia of finite element definitions

Trimmed serendipity H(curl)

Click here to read what the information on this page means.

De Rham complex families\(\left[S_{2,k}^\square\right]_{1}\) or \(\mathcal{S}^-_{k}\Lambda^{1}(\square_d)\)
Degrees\(0\leqslant k\)
where \(k\) is the Polynomial subdegree
Polynomial subdegree\(k\)
Polynomial superdegree\(k+d-1\)
Lagrange subdegree\(\operatorname{floor}((k+d)/(d+1))\)
Lagrange superdegree\(k+1\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(40)}_{k+1} \oplus \mathcal{Z}^{(41)}_{k+1}\) (quadrilateral)
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(42)}_{k+1} \oplus \mathcal{Z}^{(43)}_{k+1} \oplus \mathcal{Z}^{(44)}_{k+1} \oplus \mathcal{Z}^{(45)}_{k} \oplus \mathcal{Z}^{(46)}_{k+1}\) (hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each edge: tangential integral moments with a degree \(k\) dPc space
On each face: 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\}\)
Mappingcovariant Piola
continuityComponents tangential to facets are continuous
CategoriesVector-valued elements, H(curl) 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