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Degree 3 Hsieh–Clough–Tocher on a triangle

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In this example:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} + 111 x y^{2} + \tfrac{1369 y^{3}}{7} - \tfrac{828 y^{2}}{7} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{297 x^{3}}{7} + \tfrac{891 x^{2} y}{7} - \tfrac{858 x^{2}}{7} + \tfrac{891 x y^{2}}{7} - \tfrac{1716 x y}{7} + \tfrac{825 x}{7} + \tfrac{297 y^{3}}{7} - \tfrac{858 y^{2}}{7} + \tfrac{825 y}{7} - \tfrac{264}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{1369 x^{3}}{7} + 111 x^{2} y - \tfrac{828 x^{2}}{7} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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

\(\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} + x + \tfrac{272 y^{3}}{7} - \tfrac{108 y^{2}}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{48 x^{3}}{7} + \tfrac{24 x^{2} y}{7} + \tfrac{64 x^{2}}{7} + \tfrac{192 x y^{2}}{7} - \tfrac{208 x y}{7} + \tfrac{16 x}{7} + \tfrac{120 y^{3}}{7} - \tfrac{272 y^{2}}{7} + \tfrac{184 y}{7} - \tfrac{32}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(62 x^{2} + 203 x y - 101 x + 14 y^{2} - 21 y + 7\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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

\(\displaystyle \phi_{2} = \begin{cases} \tfrac{y \left(14 x^{2} + 203 x y - 21 x + 62 y^{2} - 101 y + 7\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{120 x^{3}}{7} + \tfrac{192 x^{2} y}{7} - \tfrac{272 x^{2}}{7} + \tfrac{24 x y^{2}}{7} - \tfrac{208 x y}{7} + \tfrac{184 x}{7} - \tfrac{48 y^{3}}{7} + \tfrac{64 y^{2}}{7} + \tfrac{16 y}{7} - \tfrac{32}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{272 x^{3}}{7} - \tfrac{108 x^{2}}{7} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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

\(\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} - 102 x y^{2} + \tfrac{377 y^{3}}{7} + \tfrac{39 y^{2}}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{1210 x^{3}}{7} - \tfrac{2958 x^{2} y}{7} + \tfrac{2913 x^{2}}{7} - \tfrac{2307 x y^{2}}{7} + \tfrac{4503 x y}{7} - \tfrac{2196 x}{7} - \tfrac{545 y^{3}}{7} + \tfrac{1590 y^{2}}{7} - \tfrac{1545 y}{7} + \tfrac{500}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{3 x^{2} \left(- 96 x - 21 y + 20\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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

\(\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} + \tfrac{51 x y^{2}}{2} - \tfrac{209 y^{3}}{14} - \tfrac{11 y^{2}}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{615 x^{3}}{14} + \tfrac{737 x^{2} y}{7} - \tfrac{732 x^{2}}{7} + \tfrac{569 x y^{2}}{7} - \tfrac{2235 x y}{14} + \tfrac{1097 x}{14} + \tfrac{265 y^{3}}{14} - \tfrac{389 y^{2}}{7} + \tfrac{761 y}{14} - \tfrac{124}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \cdot \left(141 x + 21 y - 25\right)}{14}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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

\(\displaystyle \phi_{5} = \begin{cases} \tfrac{y \left(28 x^{2} - 329 x y - 14 x - 239 y^{2} + 179 y\right)}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{503 x^{3}}{14} - \tfrac{751 x^{2} y}{7} + \tfrac{655 x^{2}}{7} - \tfrac{751 x y^{2}}{7} + \tfrac{2613 x y}{14} - \tfrac{1111 x}{14} - \tfrac{489 y^{3}}{14} + \tfrac{641 y^{2}}{7} - \tfrac{1097 y}{14} + \tfrac{152}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{15 x^{2} \left(- 15 x - 21 y + 11\right)}{14}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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

\(\displaystyle \phi_{6} = \begin{cases} \tfrac{3 y^{2} \left(- 21 x - 96 y + 20\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{545 x^{3}}{7} - \tfrac{2307 x^{2} y}{7} + \tfrac{1590 x^{2}}{7} - \tfrac{2958 x y^{2}}{7} + \tfrac{4503 x y}{7} - \tfrac{1545 x}{7} - \tfrac{1210 y^{3}}{7} + \tfrac{2913 y^{2}}{7} - \tfrac{2196 y}{7} + \tfrac{500}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{377 x^{3}}{7} - 102 x^{2} y + \tfrac{39 x^{2}}{7} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)

\(\displaystyle \phi_{7} = \begin{cases} \tfrac{15 y^{2} \left(- 21 x - 15 y + 11\right)}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{489 x^{3}}{14} - \tfrac{751 x^{2} y}{7} + \tfrac{641 x^{2}}{7} - \tfrac{751 x y^{2}}{7} + \tfrac{2613 x y}{14} - \tfrac{1097 x}{14} - \tfrac{503 y^{3}}{14} + \tfrac{655 y^{2}}{7} - \tfrac{1111 y}{14} + \tfrac{152}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(- 239 x^{2} - 329 x y + 179 x + 28 y^{2} - 14 y\right)}{14}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)

\(\displaystyle \phi_{8} = \begin{cases} \tfrac{y^{2} \cdot \left(21 x + 141 y - 25\right)}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{265 x^{3}}{14} + \tfrac{569 x^{2} y}{7} - \tfrac{389 x^{2}}{7} + \tfrac{737 x y^{2}}{7} - \tfrac{2235 x y}{14} + \tfrac{761 x}{14} + \tfrac{615 y^{3}}{14} - \tfrac{732 y^{2}}{7} + \tfrac{1097 y}{14} - \tfrac{124}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{209 x^{3}}{14} + \tfrac{51 x^{2} y}{2} - \tfrac{11 x^{2}}{14} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}\)
where \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \phi_{9} = \begin{cases} \tfrac{2 \sqrt{2} y^{2} \left(- 21 x - 15 y + 11\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{2 \sqrt{2} \left(- 41 x^{3} - 130 x^{2} y + 106 x^{2} - 130 x y^{2} + 219 x y - 89 x - 41 y^{3} + 106 y^{2} - 89 y + 24\right)}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{2 \sqrt{2} x^{2} \left(- 15 x - 21 y + 11\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{10}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}\)
where \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \phi_{10} = \begin{cases} \tfrac{4 y^{2} \left(- 21 x - 6 y + 10\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{60 x^{3}}{7} - \tfrac{96 x^{2} y}{7} + \tfrac{136 x^{2}}{7} - \tfrac{12 x y^{2}}{7} + \tfrac{104 x y}{7} - \tfrac{92 x}{7} + \tfrac{24 y^{3}}{7} - \tfrac{32 y^{2}}{7} - \tfrac{8 y}{7} + \tfrac{16}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{4 x \left(- 41 x^{2} + 7 x y + 17 x + 7 y^{2} - 7 y\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{11}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}\)
where \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \phi_{11} = \begin{cases} \tfrac{4 y \left(- 7 x^{2} - 7 x y + 7 x + 41 y^{2} - 17 y\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{24 x^{3}}{7} + \tfrac{12 x^{2} y}{7} + \tfrac{32 x^{2}}{7} + \tfrac{96 x y^{2}}{7} - \tfrac{104 x y}{7} + \tfrac{8 x}{7} + \tfrac{60 y^{3}}{7} - \tfrac{136 y^{2}}{7} + \tfrac{92 y}{7} - \tfrac{16}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{4 x^{2} \cdot \left(6 x + 21 y - 10\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)

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