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# Degree 3 vector dPc on a quadrilateral

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In this example:
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{3}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{19}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 27 x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 27 x y \left(- x - y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \frac{9 x y \left(3 x - 1\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x y \left(3 x - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{2}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{2}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{2}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \frac{9 x y \left(3 y - 1\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{2}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x y \left(3 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\end{array}\right)$$

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