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Degree 1 Guzmán–Neilan (first kind) on a triangle
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- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{\sqrt{2} y \left(- 6 x + 9 y - 4\right)}{12}\\\displaystyle \tfrac{\sqrt{2} y \left(3 y - 4\right)}{12}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{\sqrt{2} x \left(3 x - 4\right)}{12}\\\displaystyle \tfrac{\sqrt{2} x \left(9 x - 6 y - 4\right)}{12}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{\sqrt{2} \left(3 x^{2} - 6 x y - 2 x - 3 y^{2} + 4 y - 1\right)}{12}\\\displaystyle \tfrac{\sqrt{2} \left(- 3 x^{2} - 6 x y + 4 x + 3 y^{2} - 2 y - 1\right)}{12}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(3 y - 4\right)}{6}\\\displaystyle \tfrac{y}{3}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x^{2}}{2} + \tfrac{x}{3} + y^{2} - y\\\displaystyle x \left(- x + y + \tfrac{1}{3}\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{2} + 4 x y - \tfrac{7 x}{3} + \tfrac{5 y^{2}}{2} - \tfrac{10 y}{3} + \tfrac{5}{6}\\\displaystyle - x^{2} - 3 x y + \tfrac{5 x}{3} - 2 y^{2} + \tfrac{8 y}{3} - \tfrac{2}{3}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle y \left(- x + y - \tfrac{1}{3}\right)\\\displaystyle - x^{2} + x + \tfrac{y^{2}}{2} - \tfrac{y}{3}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{3}\\\displaystyle \tfrac{x \left(4 - 3 x\right)}{6}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} + 3 x y - \tfrac{8 x}{3} + y^{2} - \tfrac{5 y}{3} + \tfrac{2}{3}\\\displaystyle - \tfrac{5 x^{2}}{2} - 4 x y + \tfrac{10 x}{3} - \tfrac{3 y^{2}}{2} + \tfrac{7 y}{3} - \tfrac{5}{6}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle - x + \tfrac{3 y^{2}}{2} - 3 y + 1\\\displaystyle y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 x^{2}}{2} + 3 y^{2} - 4 y + 1\\\displaystyle x \left(- 3 x + 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{9 x^{2}}{2} + 12 x y - 8 x + \tfrac{15 y^{2}}{2} - 11 y + \tfrac{7}{2}\\\displaystyle - 3 x^{2} - 9 x y + 5 x - 6 y^{2} + 8 y - 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle - x + \tfrac{3 y^{2}}{2} - 3 y + 1\\\displaystyle y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 x^{2}}{2} + 3 y^{2} - 4 y + 1\\\displaystyle x \left(- 3 x + 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{9 x^{2}}{2} + 12 x y - 8 x + \tfrac{15 y^{2}}{2} - 11 y + \tfrac{7}{2}\\\displaystyle - 3 x^{2} - 9 x y + 5 x - 6 y^{2} + 8 y - 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(3 x - 3 y + 1\right)\\\displaystyle 3 x^{2} - 4 x - \tfrac{3 y^{2}}{2} + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x\\\displaystyle \tfrac{3 x^{2}}{2} - 3 x - y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 6 x^{2} - 9 x y + 8 x - 3 y^{2} + 5 y - 2\\\displaystyle \tfrac{15 x^{2}}{2} + 12 x y - 11 x + \tfrac{9 y^{2}}{2} - 8 y + \tfrac{7}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(3 x - 3 y + 1\right)\\\displaystyle 3 x^{2} - 4 x - \tfrac{3 y^{2}}{2} + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x\\\displaystyle \tfrac{3 x^{2}}{2} - 3 x - y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 6 x^{2} - 9 x y + 8 x - 3 y^{2} + 5 y - 2\\\displaystyle \tfrac{15 x^{2}}{2} + 12 x y - 11 x + \tfrac{9 y^{2}}{2} - 8 y + \tfrac{7}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{3 x y}{2} + x + \tfrac{9 y^{2}}{4} - y\\\displaystyle \tfrac{y \left(3 y - 4\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4}\\\displaystyle \tfrac{x \left(9 x - 6 y - 4\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} + \tfrac{x}{2} - \tfrac{3 y^{2}}{4} + y - \tfrac{1}{4}\\\displaystyle - \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} + x + \tfrac{3 y^{2}}{4} - \tfrac{y}{2} - \tfrac{1}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{3 x y}{2} + x + \tfrac{9 y^{2}}{4} - y\\\displaystyle \tfrac{y \left(3 y - 4\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4}\\\displaystyle \tfrac{x \left(9 x - 6 y - 4\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} + \tfrac{x}{2} - \tfrac{3 y^{2}}{4} + y - \tfrac{1}{4}\\\displaystyle - \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} + x + \tfrac{3 y^{2}}{4} - \tfrac{y}{2} - \tfrac{1}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{3 y \left(2 x - y\right)}{4}\\\displaystyle 3 x^{2} - 2 x - \tfrac{3 y^{2}}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4}\\\displaystyle \tfrac{x \left(15 x - 6 y - 8\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{21 x^{2}}{4} - \tfrac{21 x y}{2} + \tfrac{15 x}{2} - \tfrac{15 y^{2}}{4} + 6 y - \tfrac{9}{4}\\\displaystyle \tfrac{27 x^{2}}{4} + \tfrac{21 x y}{2} - 8 x + \tfrac{21 y^{2}}{4} - \tfrac{15 y}{2} + \tfrac{9}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{3 y \left(2 x - y\right)}{4}\\\displaystyle 3 x^{2} - 2 x - \tfrac{3 y^{2}}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4}\\\displaystyle \tfrac{x \left(15 x - 6 y - 8\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{21 x^{2}}{4} - \tfrac{21 x y}{2} + \tfrac{15 x}{2} - \tfrac{15 y^{2}}{4} + 6 y - \tfrac{9}{4}\\\displaystyle \tfrac{27 x^{2}}{4} + \tfrac{21 x y}{2} - 8 x + \tfrac{21 y^{2}}{4} - \tfrac{15 y}{2} + \tfrac{9}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 6 x + 15 y - 8\right)}{4}\\\displaystyle \tfrac{3 y^{2}}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 x^{2}}{4} + 3 y^{2} - 2 y\\\displaystyle \tfrac{3 x \left(- x + 2 y\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{21 x^{2}}{4} + \tfrac{21 x y}{2} - \tfrac{15 x}{2} + \tfrac{27 y^{2}}{4} - 8 y + \tfrac{9}{4}\\\displaystyle - \tfrac{15 x^{2}}{4} - \tfrac{21 x y}{2} + 6 x - \tfrac{21 y^{2}}{4} + \tfrac{15 y}{2} - \tfrac{9}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 6 x + 15 y - 8\right)}{4}\\\displaystyle \tfrac{3 y^{2}}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{3 x^{2}}{4} + 3 y^{2} - 2 y\\\displaystyle \tfrac{3 x \left(- x + 2 y\right)}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{21 x^{2}}{4} + \tfrac{21 x y}{2} - \tfrac{15 x}{2} + \tfrac{27 y^{2}}{4} - 8 y + \tfrac{9}{4}\\\displaystyle - \tfrac{15 x^{2}}{4} - \tfrac{21 x y}{2} + 6 x - \tfrac{21 y^{2}}{4} + \tfrac{15 y}{2} - \tfrac{9}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 6 x + 9 y - 4\right)}{4}\\\displaystyle \tfrac{3 y^{2}}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(3 x - 4\right)}{4}\\\displaystyle \tfrac{9 x^{2}}{4} - \tfrac{3 x y}{2} - x + y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} - \tfrac{x}{2} - \tfrac{3 y^{2}}{4} + y - \tfrac{1}{4}\\\displaystyle - \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} + x + \tfrac{3 y^{2}}{4} + \tfrac{y}{2} - \tfrac{1}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 6 x + 9 y - 4\right)}{4}\\\displaystyle \tfrac{3 y^{2}}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(3 x - 4\right)}{4}\\\displaystyle \tfrac{9 x^{2}}{4} - \tfrac{3 x y}{2} - x + y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} - \tfrac{x}{2} - \tfrac{3 y^{2}}{4} + y - \tfrac{1}{4}\\\displaystyle - \tfrac{3 x^{2}}{4} - \tfrac{3 x y}{2} + x + \tfrac{3 y^{2}}{4} + \tfrac{y}{2} - \tfrac{1}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{6} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 6 x + 9 y - 4\right)}{2}\\\displaystyle \tfrac{y \left(3 y - 4\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(3 x - 4\right)}{2}\\\displaystyle \tfrac{x \left(9 x - 6 y - 4\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{2} - 3 x y - x - \tfrac{3 y^{2}}{2} + 2 y - \tfrac{1}{2}\\\displaystyle - \tfrac{3 x^{2}}{2} - 3 x y + 2 x + \tfrac{3 y^{2}}{2} - y - \tfrac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{6} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 6 x + 9 y - 4\right)}{2}\\\displaystyle \tfrac{y \left(3 y - 4\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(3 x - 4\right)}{2}\\\displaystyle \tfrac{x \left(9 x - 6 y - 4\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{3 x^{2}}{2} - 3 x y - x - \tfrac{3 y^{2}}{2} + 2 y - \tfrac{1}{2}\\\displaystyle - \tfrac{3 x^{2}}{2} - 3 x y + 2 x + \tfrac{3 y^{2}}{2} - y - \tfrac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(3 y - 4\right)\\\displaystyle 2 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 3 x^{2} + 2 x + 6 y^{2} - 6 y\\\displaystyle 2 x \left(- 3 x + 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 9 x^{2} + 24 x y - 14 x + 15 y^{2} - 20 y + 5\\\displaystyle - 6 x^{2} - 18 x y + 10 x - 12 y^{2} + 16 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(3 y - 4\right)\\\displaystyle 2 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 3 x^{2} + 2 x + 6 y^{2} - 6 y\\\displaystyle 2 x \left(- 3 x + 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 9 x^{2} + 24 x y - 14 x + 15 y^{2} - 20 y + 5\\\displaystyle - 6 x^{2} - 18 x y + 10 x - 12 y^{2} + 16 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(- 3 x + 3 y - 1\right)\\\displaystyle - 6 x^{2} + 6 x + 3 y^{2} - 2 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x\\\displaystyle x \left(4 - 3 x\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 12 x^{2} + 18 x y - 16 x + 6 y^{2} - 10 y + 4\\\displaystyle - 15 x^{2} - 24 x y + 20 x - 9 y^{2} + 14 y - 5\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(- 3 x + 3 y - 1\right)\\\displaystyle - 6 x^{2} + 6 x + 3 y^{2} - 2 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x\\\displaystyle x \left(4 - 3 x\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 12 x^{2} + 18 x y - 16 x + 6 y^{2} - 10 y + 4\\\displaystyle - 15 x^{2} - 24 x y + 20 x - 9 y^{2} + 14 y - 5\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.