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# Degree 2 Lagrange on a tetrahedron

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In this example:
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$, $$z$$, $$x z$$, $$y z$$, $$z^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

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

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

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

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

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

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

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

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

$$\displaystyle \phi_{4} = 4 y z$$

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

$$\displaystyle \phi_{5} = 4 x z$$

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

$$\displaystyle \phi_{6} = 4 x y$$

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

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(0,\tfrac{1}{2},0)$$

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

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{9}:v\mapsto v(\tfrac{1}{2},0,0)$$

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

This DOF is associated with edge 5 of the reference element.