an encyclopedia of finite element definitions

# Degree 2 Lagrange on a prism

◀ Back to Lagrange definition page In this example:
• $$R$$ is the reference prism. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$, $$z$$, $$x z$$, $$x^{2} z$$, $$y z$$, $$x y z$$, $$y^{2} z$$, $$z^{2}$$, $$x z^{2}$$, $$x^{2} z^{2}$$, $$y z^{2}$$, $$x y z^{2}$$, $$y^{2} z^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{17}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

$$\displaystyle \phi_{0} = 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 8 x y z^{2} - 12 x y z + 4 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1$$

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

$$\displaystyle \phi_{1} = x \left(4 x z^{2} - 6 x z + 2 x - 2 z^{2} + 3 z - 1\right)$$

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

$$\displaystyle \phi_{2} = y \left(4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)$$

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

$$\displaystyle \phi_{3} = z \left(4 x^{2} z - 2 x^{2} + 8 x y z - 4 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)$$

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

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

This DOF is associated with vertex 4 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(0,1,1)$$

$$\displaystyle \phi_{5} = y z \left(4 y z - 2 y - 2 z + 1\right)$$

This DOF is associated with vertex 5 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(\tfrac{1}{2},0,0)$$

$$\displaystyle \phi_{6} = 4 x \left(- 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{7}:v\mapsto v(0,\tfrac{1}{2},0)$$

$$\displaystyle \phi_{7} = 4 y \left(- 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(0,0,\tfrac{1}{2})$$

$$\displaystyle \phi_{8} = 4 z \left(- 2 x^{2} z + 2 x^{2} - 4 x y z + 4 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},0)$$

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{10}:v\mapsto v(1,0,\tfrac{1}{2})$$

$$\displaystyle \phi_{10} = 4 x z \left(- 2 x z + 2 x + z - 1\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{11}:v\mapsto v(0,1,\tfrac{1}{2})$$

$$\displaystyle \phi_{11} = 4 y z \left(- 2 y z + 2 y + z - 1\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{12}:v\mapsto v(\tfrac{1}{2},0,1)$$

$$\displaystyle \phi_{12} = 4 x z \left(- 2 x z + x - 2 y z + y + 2 z - 1\right)$$

This DOF is associated with edge 6 of the reference element.
$$\displaystyle l_{13}:v\mapsto v(0,\tfrac{1}{2},1)$$

$$\displaystyle \phi_{13} = 4 y z \left(- 2 x z + x - 2 y z + y + 2 z - 1\right)$$

This DOF is associated with edge 7 of the reference element.
$$\displaystyle l_{14}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},1)$$

$$\displaystyle \phi_{14} = 4 x y z \left(2 z - 1\right)$$

This DOF is associated with edge 8 of the reference element.
$$\displaystyle l_{15}:v\mapsto v(\tfrac{1}{2},0,\tfrac{1}{2})$$

$$\displaystyle \phi_{15} = 16 x z \left(x z - x + y z - y - z + 1\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{16}:v\mapsto v(0,\tfrac{1}{2},\tfrac{1}{2})$$

$$\displaystyle \phi_{16} = 16 y z \left(x z - x + y z - y - z + 1\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{17}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$$

$$\displaystyle \phi_{17} = 16 x y z \left(1 - z\right)$$

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