an encyclopedia of finite element definitions

# Transition

 Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron DOFs On each vertex: point evaluations On each edge: point evaluations On each face: point evaluations On each volume: point evaluations Notes This element is used to bridge the gap between Lagrange elements of different orders Categories Scalar-valued elements

## Implementations

 Symfem "transition"↓ Show Symfem examples ↓

## Examples

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

$$\displaystyle \phi_{0} = - x - y + 1$$

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

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

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

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

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

$$\displaystyle \phi_{3} = 4 x y$$

This DOF is associated with edge 0 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: $$1$$, $$x$$, $$y$$, $$x y \left(1 - y\right)$$, $$x y^{2}$$, $$y \left(- x - y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

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

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

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

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

$$\displaystyle \phi_{2} = \frac{y \left(- 9 x y + 4 x + 4 y - 2\right)}{2}$$

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

$$\displaystyle \phi_{3} = \frac{9 x y \left(2 - 3 y\right)}{2}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})$$

$$\displaystyle \phi_{4} = \frac{9 x y \left(3 y - 1\right)}{2}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{2})$$

$$\displaystyle \phi_{5} = 4 y \left(- x - y + 1\right)$$

This DOF is associated with edge 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: $$1$$, $$x$$, $$y$$, $$x y \left(- x - y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1$$

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

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

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

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

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

$$\displaystyle \phi_{3} = 27 x y \left(- x - y + 1\right)$$

This DOF is associated with face 0 of the reference element.

## DefElement stats

 Element added 12 July 2021 Element last updated 12 July 2021