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# Tiniest tensor H(curl)

 Alternative names TNT H(curl) Cockburn–fu names $$\left[S_{3,k}^\square\right]_{1}$$ Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(44)}_{k}$$ (quadrilateral) $$\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(45)}_{k} \oplus \mathcal{Z}^{(46)}_{k} \oplus \mathcal{Z}^{(47)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each edge: tangent integral moments with an order $$k$$ 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 an order $$k$$ 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 an order $$k$$ 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 an order $$k$$ 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 an order $$k$$ 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 tandential to facets are continuous Categories Vector-valued elements, H(curl) conforming elements

## Implementations

 Symfem "TNTcurl" (quadrilateral, hexahedron)↓ Show Symfem examples ↓

## Examples

 quadrilateralorder 1 (click to view basis functions) quadrilateralorder 2 (click to view basis functions) quadrilateralorder 3 (click to view basis functions) hexahedronorder 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 02 August 2022