Trimmed serendipity H(curl)

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De Rham complex families	\(\left[S_{2,k}^\square\right]_{1}\) or \(\mathcal{S}^-_{k}\Lambda^{1}(\square_d)\)
Degrees	\(1\leqslant k\) where \(k\) is the Lagrange superdegree
Polynomial subdegree	\(k-1\)
Polynomial superdegree	\(k+d-2\)
Lagrange subdegree	\(\operatorname{floor}((k+d-1)/(d+1))\)
Lagrange superdegree	\(k\)
Reference elements	quadrilateral, hexahedron
Polynomial set	\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(40)}_{k} \oplus \mathcal{Z}^{(41)}_{k}\) (quadrilateral) \(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(42)}_{k} \oplus \mathcal{Z}^{(43)}_{k} \oplus \mathcal{Z}^{(44)}_{k} \oplus \mathcal{Z}^{(45)}_{k} \oplus \mathcal{Z}^{(46)}_{k}\) (hexahedron) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(40)}_k=\operatorname{span}\left\{\left(\begin{array}{c}y\\-x\end{array}\right)p\middle\|p\in\hat{\mathcal{P}}_{k-1}\right\}\) \(\hat{\mathcal{P}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i=k\right\}=\mathcal{P}_k\setminus\mathcal{P}_{k-1}\) \(\mathcal{Z}^{(41)}_k=\operatorname{span}\left\{\left(\begin{array}{c}y^k\\kxy^{k-1}\end{array}\right),\left(\begin{array}{c}kx^{k-1}y\\x^k\end{array}\right)\right\}\) \(\mathcal{Z}^{(42)}_k=\operatorname{span}\left\{\boldsymbol{x}\times p\middle\|p\in\hat{\mathcal{P}}_{k-1}\right\}\) \(\mathcal{Z}^{(43)}_k=\operatorname{span}\left\{\nabla p\middle\|p\in\{xyz^k,xy^kz,x^kyz\}\cup\{xy^az^{k-a}\|a=0,1,...,k\}\cup\{x^ayz^{k-a}\|a=0,1,...,k\}\cup\{x^ay^{k-a}z\|a=0,1,...,k\}\right\}\) \(\mathcal{Z}^{(44)}_k=\operatorname{span}\left\{\left(\left(\begin{array}{c}-y\\x\\0\end{array}\right)xy^az^{k-1-a}\right)\middle\|a=0,1,...,a-1\right\}\) \(\mathcal{Z}^{(45)}_k=\operatorname{span}\left\{\left(\left(\begin{array}{c}-z\\0\\x\end{array}\right)yx^az^{k-1-a}\right)\middle\|a=0,1,...,a-1\right\}\) \(\mathcal{Z}^{(46)}_k=\operatorname{span}\left\{\left(\left(\begin{array}{c}-y\\x\\0\end{array}\right)zx^ay^{k-1-a}\right)\middle\|a=0,1,...,a-1\right\}\)
DOFs	On each edge: tangential integral moments with a degree \(k-1\) dPc space On each face: integral moments with a degree \(k-3\) vector dPc space, and integral moments with \(\left\{\nabla(p)\middle\|p\text{ is a degree \(k-1\) monomial}\right\}\)
Mapping	covariant Piola
continuity	Components tangential to facets are continuous
Categories	Vector-valued elements, H(curl) conforming elements

Implementations

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

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FIAT

FIAT.TrimmedSerendipityCurl
↓ Show FIAT examples ↓ This implementation is correct for all the examples below.

Symfem

"TScurl"
↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.

Examples

quadrilateral degree 1	(click to view basis functions)
quadrilateral degree 2	(click to view basis functions)
quadrilateral degree 3	(click to view basis functions)
hexahedron degree 1	(click to view basis functions)
hexahedron degree 2	(click to view basis functions)
hexahedron degree 3	(click to view basis functions)

References

Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM Journal of Numerical Analysis 55(4), 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]
Gillette, Andrew and Kloefkorn, Tyler. Trimmed serendipity finite element differential forms, Mathematics of Computation 88, 583–606, 2019. [DOI: 10.1090/mcom/3354] [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	07 October 2021
Element last updated	16 October 2024