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Tiniest tensor H(curl)

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Alternative names	TNT H(curl)
De Rham complex families	\(\left[S_{3,k}^\square\right]_{1}\)
Degrees	\(1\leqslant k\) where \(k\) is the Polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(dk + 1\)
Lagrange subdegree	\(k\)
Lagrange superdegree	\(k+1\)
Reference elements	quadrilateral, hexahedron
Polynomial set	\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(34)}_{k}\) (quadrilateral) \(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(35)}_{k} \oplus \mathcal{Z}^{(36)}_{k} \oplus \mathcal{Z}^{(37)}_{k}\) (hexahedron) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(34)}_k=\left\{\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\end{array}\right)\right\}] \) where \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\) \(\tilde{B}_k\) is \((\tilde{P}_{k+2}-\tilde{P}_k)/(4k+6)\) \(\mathcal{Z}^{(35)}_k=\left\{f\boldsymbol{g}\middle\|f\in\{x,1-x\},\boldsymbol{g}\in\left\{\left(\begin{array}{c}\tilde{P}_k(y)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(y)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(y)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(y)\tilde{P}_{k}(0)\end{array}\right),\right\}\right\} \) where \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\) \(\tilde{B}_k\) is \((\tilde{P}_{k+2}-\tilde{P}_k)/(4k+6)\) \(\mathcal{Z}^{(36)}_k=\left\{f\boldsymbol{g}\middle\|f\in\{y,1-y\},\boldsymbol{g}\in\left\{\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\end{array}\right),\right\}\right\}\) \(\mathcal{Z}^{(37)}_k=\left\{f\boldsymbol{g}\middle\|f\in\{z,1-z\},\boldsymbol{g}\in\left\{\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\end{array}\right),\right\}\right\}\)
DOFs	On each edge: tangent integral moments with an degree \(k\) Lagrange space On each face: integral moments with \(\nabla f\times \boldsymbol{n}\) (where \(\boldsymbol{n}\) is a unit vector normal to the face) for each \(f\) in a degree \(k\) Lagrange space, and integral moments with \(\nabla\times\boldsymbol{f}\times \boldsymbol{n}\) (where \(\boldsymbol{n}\) is a unit vector normal to the face) for each \(\boldsymbol{f}\) in a degree \(k\) vector Lagrange space such that \(\nabla\cdot\boldsymbol{f}=0\) and the normal trace of \(\boldsymbol{f}\) on the edges of the face is 0 On each volume: integral moments with \(\nabla\times\boldsymbol{f}\) for each \(\boldsymbol{f}\) in a degree \(k\) vector Lagrange space such that the tangential trace of \(\boldsymbol{f}\) on the faces of the volume is 0, and integral moments with \(\nabla f\) for each \(f\) in a degree \(k\) Lagrange space such that the trace of \(f\) on the faces of the volume is 0
Number of DOFs	quadrilateral: \(2(k+1)^2 + 3\) hexahedron: \(3(k+1)^3 + 18\)
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 .↓ Show implementation detail ↓

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)

References

Cockburn, Bernardo and Qiu, Weifeng. Commuting diagrams for the TNT elements on cubes, Mathematics of Computation 83, 603–633, 2014. [DOI: 10.1090/S0025-5718-2013-02729-9] [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	24 October 2021
Element last updated	27 September 2024