an encyclopedia of finite element definitions

# Hermite

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 Orders $$k=3$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations, and point evaluations of derivatives in coordinate directions On each face: point evaluations at midpoints Number of DOFs interval: $$4$$triangle: $$10$$tetrahedron: $$20$$ Mapping identity continuity Function values are continuous. Notes The derivatives of the basis functions are continuous between cells at the vertices of the element Categories Scalar-valued elements

## Implementations

 Basix basix.ElementFamily.Hermite↓ Show Basix examples ↓ This implementation is correct for all the examples below. FIAT FIAT.CubicHermite↓ Show FIAT examples ↓ This implementation is correct for all the examples below. Symfem "Hermite"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "Hermite"↓ Show (legacy) UFL examples ↓

## Examples

 intervalorder 3 (click to view basis functions) triangleorder 3 (click to view basis functions) tetrahedronorder 3 (click to view basis functions)

## References

• Ciarlet, Philippe G. and Raviart, Pierre-Arnaud. Interpolation theory over curved elements, with applications to finite element methods, Computer Methods in Applied Mechanics and Engineering 1(2), 217–249, 1972. [DOI: 10.1016/0045-7825(72)90006-0] [BibTeX]

## DefElement stats

 Element added 09 January 2021 Element last updated 16 September 2023