an encyclopedia of finite element definitions

# Hermite

 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: $$20$$ Categories Scalar-valued elements

## Implementations

 Symfem string "Hermite"↓ Show Symfem examples ↓ UFL string "Hermite"↓ Show UFL examples ↓

## Examples

interval
order 3
triangle
order 3
tetrahedron
order 3
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0)$$

$$\displaystyle \phi_{0} = 2 x^{3} - 3 x^{2} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v'(0)$$

$$\displaystyle \phi_{1} = x \left(x^{2} - 2 x + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(1)$$

$$\displaystyle \phi_{2} = x^{2} \left(3 - 2 x\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{3}:v\mapsto v'(1)$$

$$\displaystyle \phi_{3} = x^{2} \left(x - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 2 x^{3} + 13 x^{2} y - 3 x^{2} + 13 x y^{2} - 13 x y + 2 y^{3} - 3 y^{2} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(0,0)$$

$$\displaystyle \phi_{1} = x \left(x^{2} + 3 x y - 2 x + 2 y^{2} - 3 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(0,0)$$

$$\displaystyle \phi_{2} = y \left(2 x^{2} + 3 x y - 3 x + y^{2} - 2 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{3} = x \left(- 2 x^{2} + 7 x y + 3 x + 7 y^{2} - 7 y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(1,0)$$

$$\displaystyle \phi_{4} = x \left(x^{2} - 2 x y - x - 2 y^{2} + 2 y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(1,0)$$

$$\displaystyle \phi_{5} = x y \left(2 x + y - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{6} = y \left(7 x^{2} + 7 x y - 7 x - 2 y^{2} + 3 y\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(0,1)$$

$$\displaystyle \phi_{7} = x y \left(x + 2 y - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(0,1)$$

$$\displaystyle \phi_{8} = y \left(- 2 x^{2} - 2 x y + 2 x + y^{2} - y\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$, $$z$$, $$x z$$, $$x^{2} z$$, $$y z$$, $$x y z$$, $$y^{2} z$$, $$z^{2}$$, $$x z^{2}$$, $$y z^{2}$$, $$z^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{19}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

$$\displaystyle \phi_{0} = 2 x^{3} + 13 x^{2} y + 13 x^{2} z - 3 x^{2} + 13 x y^{2} + 33 x y z - 13 x y + 13 x z^{2} - 13 x z + 2 y^{3} + 13 y^{2} z - 3 y^{2} + 13 y z^{2} - 13 y z + 2 z^{3} - 3 z^{2} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(0,0,0)$$

$$\displaystyle \phi_{1} = x \left(x^{2} + 3 x y + 3 x z - 2 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(0,0,0)$$

$$\displaystyle \phi_{2} = y \left(2 x^{2} + 3 x y + 4 x z - 3 x + y^{2} + 3 y z - 2 y + 2 z^{2} - 3 z + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto\frac{\partial}{\partial z}\nabla{v}(0,0,0)$$

$$\displaystyle \phi_{3} = z \left(2 x^{2} + 4 x y + 3 x z - 3 x + 2 y^{2} + 3 y z - 3 y + z^{2} - 2 z + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(1,0,0)$$

$$\displaystyle \phi_{4} = x \left(- 2 x^{2} + 7 x y + 7 x z + 3 x + 7 y^{2} + 7 y z - 7 y + 7 z^{2} - 7 z\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(1,0,0)$$

$$\displaystyle \phi_{5} = x \left(x^{2} - 2 x y - 2 x z - x - 2 y^{2} - 2 y z + 2 y - 2 z^{2} + 2 z\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(1,0,0)$$

$$\displaystyle \phi_{6} = x y \left(2 x + y - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial z}\nabla{v}(1,0,0)$$

$$\displaystyle \phi_{7} = x z \left(2 x + z - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(0,1,0)$$

$$\displaystyle \phi_{8} = y \left(7 x^{2} + 7 x y + 7 x z - 7 x - 2 y^{2} + 7 y z + 3 y + 7 z^{2} - 7 z\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(0,1,0)$$

$$\displaystyle \phi_{9} = x y \left(x + 2 y - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{10}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(0,1,0)$$

$$\displaystyle \phi_{10} = y \left(- 2 x^{2} - 2 x y - 2 x z + 2 x + y^{2} - 2 y z - y - 2 z^{2} + 2 z\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{11}:v\mapsto\frac{\partial}{\partial z}\nabla{v}(0,1,0)$$

$$\displaystyle \phi_{11} = y z \left(2 y + z - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{12}:v\mapsto v(0,0,1)$$

$$\displaystyle \phi_{12} = z \left(7 x^{2} + 7 x y + 7 x z - 7 x + 7 y^{2} + 7 y z - 7 y - 2 z^{2} + 3 z\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{13}:v\mapsto\frac{\partial}{\partial x}\nabla{v}(0,0,1)$$

$$\displaystyle \phi_{13} = x z \left(x + 2 z - 1\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{14}:v\mapsto\frac{\partial}{\partial y}\nabla{v}(0,0,1)$$

$$\displaystyle \phi_{14} = y z \left(y + 2 z - 1\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{15}:v\mapsto\frac{\partial}{\partial z}\nabla{v}(0,0,1)$$

$$\displaystyle \phi_{15} = z \left(- 2 x^{2} - 2 x y - 2 x z + 2 x - 2 y^{2} - 2 y z + 2 y + z^{2} - z\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{16}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{16} = 27 x y z$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:v\mapsto v(0,\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{17} = 27 y z \left(- x - y - z + 1\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{18}:v\mapsto v(\tfrac{1}{3},0,\tfrac{1}{3})$$

$$\displaystyle \phi_{18} = 27 x z \left(- x - y - z + 1\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{19}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3},0)$$

$$\displaystyle \phi_{19} = 27 x y \left(- x - y - z + 1\right)$$

This DOF is associated with face 3 of the reference element.

## References

• Ciarlet, P. G. and Raviart, P.-A. 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]