an encyclopedia of finite element definitions

# Bernardi–Raugel

 Abbreviated names BR Orders $$k=1$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(4)}_{k}$$ (triangle) $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(5)}_{k}$$ (tetrahedron, $$k=1$$) $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(6)}_{k}$$ (tetrahedron, $$k=2$$) ↓ Show polynomial set definitions ↓ DOFs On 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 DOFs triangle: $$9$$tetrahedron: $$\begin{cases}16&k=1\\37&k=2\end{cases}$$ Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "Bernardi-Raugel"↓ Show Symfem examples ↓

## Examples

 triangleorder 1 (click to view basis functions) tetrahedronorder 1 (click to view basis functions) tetrahedronorder 2 (click to view basis functions)

## References

• Bernardi, Christine and Raugel, Genevivève. Analysis of some finite elements for the Stokes problem, Mathematics of Computation 44, 71–79, 1985. [DOI: 10.1090/S0025-5718-1985-0771031-7] [BibTeX]

## DefElement stats

 Element added 19 April 2021 Element last updated 02 August 2022