Serendipity H(curl)

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Alternative names	Brezzi–Douglas–Marini cubical H(curl) (quadrilateral), Arnold–Awanou H(curl) (hexahedron)
De Rham complex families	\(\left[S_{1,k}^\square\right]_{1}\) or \(\mathcal{S}_{k}\Lambda^{1}(\square_d)\)
Abbreviated names	BDMce (quadrilateral), AAe (hexahedron)
Degrees	\(1\leqslant k\) where \(k\) is the polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k + d - 1\)
Lagrange subdegree	quadrilateral: \(\operatorname{floor}(k/d)\) hexahedron: \(1\) (degree=2), \(\operatorname{floor}(k/d)\) (otherwise)
Lagrange superdegree	\(k+1\)
Reference cells	quadrilateral, hexahedron
Finite dimensional space	\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(26)}_{k}\) (quadrilateral) \(\mathcal{P}_{k}^d \oplus \mathcal{A}_{k-1} \oplus \mathcal{Z}^{(27)}_{k}\) (hexahedron) ↓ Show set definitions ↓↑ Hide set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(26)}_k=\operatorname{span}\left\{\left(\begin{array}{c}(k+1)x^ky\\-x^{k+1}\end{array}\right),\left(\begin{array}{c}y^{k+1}\\-(k+1)xy^k\end{array}\right)\right\}\) \(\mathcal{A}_k=\operatorname{span}\bigcup_{p\in\mathcal{P}_k(\mathbb{R}^{d-1})}\left\{\left(\begin{array}{c}0\\xzp(y,z)\\-xyp(y,z)\end{array}\right),\left(\begin{array}{c}yzp(x,z)\\0\\-xyp(x,z)\end{array}\right),\left(\begin{array}{c}yzp(x,y)\\-xzp(x,y)\\0\end{array}\right)\right\}\) \(\mathcal{Z}^{(27)}_k=\left\{\nabla p\middle\|p\in\mathcal{X}_{k+1}\right\}\) \(\mathcal{X}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|k<\sum_{i=1}^da_i\leqslant k+\#\left\{i\in\{1,\dots,d\}\middle\| a_i=1\right\}\right\}\)
DOFs	On each edge: tangent integral moments with a degree \(k\) dPc space On each face: integral moments with a degree \(k-2\) vector dPc space On each volume: integral moments with a degree \(k-4\) vector dPc space
Number of DOFs	quadrilateral: \(k^2+3k+4\) (A014206) hexahedron: \(\begin{cases}6(k^2+k+2)&k=1,2,3\\k(k+1)(k-1)/2 + 3k^2 + 12k + 9&k > 3\end{cases}\)
Mapping	covariant Piola
continuity	Components tangential to facets are continuous
Categories	Vector-valued elements, H(curl) conforming elements

Implementations

This element is implemented in Symfem and (legacy) UFL.↓ Show implementation detail ↓

Examples

quadrilateral degree 1	(click to view basis functions)
quadrilateral degree 2	(click to view basis functions)
hexahedron degree 1	(click to view basis functions)
hexahedron degree 2	(click to view basis functions)

References

Arnold, Douglas N. and Awanou, Gerard. Finite element differential forms on cubical meshes, Mathematics of computation 83, 1551–5170, 2014. [DOI: 10.1090/S0025-5718-2013-02783-4] [BibTeX]
Arnold, Douglas N. and Logg, Anders. Periodic table of the finite elements, SIAM News 47, 2014. [www.siam.org/publications/siam-news/issues/volume-47-number-09-november-2014/] [BibTeX]
Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM journal on numerical analysis 55, 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]

DefElement stats

Element added	31 December 2020
Element last updated	04 June 2025