an encyclopedia of finite element definitions

# Trimmed serendipity H(div)

 Exterior calculus names $$\mathcal{S}^-_{k}\Lambda^{d-1}(\square_d)$$ Cockburn–fu names $$\left[S_{2,k}^\square\right]_{d-1}$$ Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(5)}_{k} \oplus \mathcal{Z}^{(6)}_{k}$$ (quadrilateral) $$\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(5)}_{k} \oplus \mathcal{Z}^{(7)}_{k} \oplus \mathcal{Z}^{(8)}_{k} \oplus \mathcal{Z}^{(9)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k-1$$ dPc space On the interior of the reference element: integral moments with an order $$k-3$$ vector dPc space, and integral moments with $$\left\{\nabla(p)\middle|p\text{ is an order \(k-1$$ monomial}\right\}\) Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "TSdiv"↓ 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) hexahedronorder 2 (click to view basis functions) hexahedronorder 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. [sinews.siam.org/Details-Page/periodic-table-of-the-finite-elements] [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 02 August 2022