an encyclopedia of finite element definitions

Bernardi–Raugel

Click here to read what the information on this page means.

Abbreviated namesBR
Orders\(k=1\)
Reference elementstriangle, tetrahedron
Polynomial set\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(48)}_{k}\) (triangle)
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(49)}_{k}\) (tetrahedron, \(k=1\))
\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(50)}_{k}\) (tetrahedron, \(k=2\))
↓ Show polynomial set definitions ↓
DOFsOn each edge: (if \(k>1\)) point evaluations in tangential directions at midpoints
On each facet: point evaluations in normal directions at vertices,
normal integral moments with an order \(k-1\) Lagrange space, and (if \(k>1\)) point evaluations in normal directions at midpoints of edges
On the interior of the reference element: integral moments of the divergence with an order \(0\) vector Lagrange space
Number of DOFstriangle: \(9\)
tetrahedron: \(\begin{cases}16&k=1\\37&k=2\end{cases}\)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"Bernardi-Raugel"
↓ Show Symfem examples ↓

Examples

triangle
order 1

(click to view basis functions)
tetrahedron
order 1

(click to view basis functions)
tetrahedron
order 2

(click to view basis functions)

References

DefElement stats

Element added19 April 2021
Element last updated02 August 2022