an encyclopedia of finite element definitions

# Tiniest tensor

 Alternative names TNT Cockburn–fu names $$\left[S_{3,k}^\square\right]_{0}$$ Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{Q}_{k} \oplus \mathcal{Z}^{(30)}_{k}$$ (quadrilateral) $$\mathcal{Q}_{k} \oplus \mathcal{Z}^{(31)}_{k} \oplus \mathcal{Z}^{(32)}_{k} \oplus \mathcal{Z}^{(33)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: integral moments with $$\frac{\partial}{\partial x}f$$ for each $$f$$ in an order $$k$$ Lagrange space On each face: integral moments with $$\Delta f$$ for each $$f$$ in an order $$k$$ Lagrange space such that the trace of $$f$$ on the edges of the face is 0 On each volume: integral moments of gradient 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: $$(k+1)^2 + 4$$hexahedron: $$(k+1)^3 + 12$$ Mapping identity continuity Function values are continuous. Categories Scalar-valued elements

## Implementations

 Symfem "TNT" (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