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# Degree 2 vector Lagrange on a triangle

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In this example:
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\end{array}\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\end{array}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(2 y - 1\right)\end{array}\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 4 x y\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 4 y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 4 x \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(- x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.