an encyclopedia of finite element definitions

# Q H(div)

 Alternative names Raviart–Thomas cubical H(div) (quadrilateral), Nédélec cubical H(div) (hexahedron) De Rham complex families $$\left[S_{4,k}^\square\right]_{d-1}$$ / $$\mathcal{Q}^-_{k}\Lambda^{d-1}(\square_d)$$ Abbreviated names RTcf (quadrilateral), Ncf (hexahedron) Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(23)}_{k}$$ (quadrilateral) $$\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(23)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k-1$$ Lagrange space On the interior of the reference element: integral moments with an order $$k-1$$ Nédélec (first kind) space Number of DOFs quadrilateral: $$2k(k+1)$$ (A046092)hexahedron: $$3k^2(k+1)$$ (A270205) Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "Qdiv"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "RTCF" (quadrilateral)"NCF" (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)

## DefElement stats

 Element added 31 December 2020 Element last updated 16 September 2023