an encyclopedia of finite element definitions

# Serendipity H(div)

 Alternative names Brezzi–Douglas–Marini cubical H(div) (quadrilateral), Arnold–Awanou H(div) (hexahedron) De Rham complex families $$\left[S_{1,k}^\square\right]_{d-1}$$ / $$\mathcal{S}_{k}\Lambda^{d-1}(\square_d)$$ Abbreviated names BDMcf (quadrilateral), AAf (hexahedron) Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(29)}_{k}$$ (quadrilateral) $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(30)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k$$ dPc space On the interior of the reference element: integral moments with an order $$k-2$$ vector dPc space Number of DOFs quadrilateral: $$k^2+3k+4$$ (A014206)hexahedron: $$(k+1)(k^2+5k+12)/2$$ Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "Sdiv"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "BDMCF" (quadrilateral)"AAF" (hexahedron)↓ Show (legacy) UFL examples ↓

## Examples

 quadrilateralorder 1 (click to view basis functions) quadrilateralorder 2 (click to view basis functions) hexahedronorder 1 (click to view basis functions) hexahedronorder 2 (click to view basis functions)
• Arnold, Douglas N. and Awanou, Gerard. Finite element differential forms on cubical meshes, Mathematics of computation 83, 1551–5170, 2014. [BibTeX]
• Brezzi, Franco, Douglas, Jim, and Marini, L. Donatella. Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik 47, 217–235, 1985. [DOI: 10.1007/BF01389710] [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 30 December 2020 Element last updated 16 September 2023