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Degree 2 Alfeld–Sorokina 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 4 x + 12 y^{2} - 4 y\\\displaystyle 2 y \left(3 x - 2\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2} + 2 x + 4 y - 1\\\displaystyle - 3 x^{2} + 4 x - 2 y - 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y + 3 y^{2}\\\displaystyle 9 x^{2} - 4 x - 3 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 144 x y + 10 x + 12 y^{2} - 10 y\\\displaystyle 2 y \left(- 15 x + 18 y - 5\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 69 x^{2} + 104 x + 46 y - 25\\\displaystyle 24 x^{2} - 26 x + 4 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 39 x^{2} + 96 x y + 21 y^{2}\\\displaystyle - 10 x + 6 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle - 14 x + 12 y^{2} + 14 y\\\displaystyle 2 y \left(21 x - 54 y + 7\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2} - 16 x + 22 y - 1\\\displaystyle 24 x^{2} + 144 x y - 50 x - 92 y + 26\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 33 x^{2} + 24 x y + 21 y^{2}\\\displaystyle 14 x - 66 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
- \(\mathcal{L}=\{l_0,...,l_{14}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 x^{2} + 3 x y - 3 x - 4 y^{2} + 1\\\displaystyle y \left(- 4 x - 3 y + 3\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x^{2} + 5 x y - 3 x + 4 y^{2} - 6 y + 2\\\displaystyle 2 x^{2} + x y - 3 x - y^{2} + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x^{2} + 2 y^{2} - 3 y + 1\\\displaystyle x \left(- 6 x - y + 3\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 x^{2} + 3 x y - 3 x - 4 y^{2} + 1\\\displaystyle y \left(- 4 x - 3 y + 3\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x^{2} + 5 x y - 3 x + 4 y^{2} - 6 y + 2\\\displaystyle 2 x^{2} + x y - 3 x - y^{2} + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x^{2} + 2 y^{2} - 3 y + 1\\\displaystyle x \left(- 6 x - y + 3\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(- x - 6 y + 3\right)\\\displaystyle 2 x^{2} - 3 x - y^{2} + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x^{2} + x y + 2 y^{2} - 3 y + 1\\\displaystyle 4 x^{2} + 5 x y - 6 x + y^{2} - 3 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x \left(- 3 x - 4 y + 3\right)\\\displaystyle - 4 x^{2} + 3 x y + 2 y^{2} - 3 y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(- x - 6 y + 3\right)\\\displaystyle 2 x^{2} - 3 x - y^{2} + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x^{2} + x y + 2 y^{2} - 3 y + 1\\\displaystyle 4 x^{2} + 5 x y - 6 x + y^{2} - 3 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x \left(- 3 x - 4 y + 3\right)\\\displaystyle - 4 x^{2} + 3 x y + 2 y^{2} - 3 y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\nabla a\cdot\boldsymbol{v}(0,0)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 2 x - 3 y + 2\right)}{2}\\\displaystyle \tfrac{y \left(- 2 x - 3 y + 2\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x^{2}}{2} + x y - x + \tfrac{y^{2}}{2} - y + \tfrac{1}{2}\\\displaystyle \tfrac{x^{2}}{2} + x y - x + \tfrac{y^{2}}{2} - y + \tfrac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 3 x - 2 y + 2\right)}{2}\\\displaystyle \tfrac{x \left(- 3 x - 2 y + 2\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y \left(- 2 x - 3 y + 2\right)}{2}\\\displaystyle \tfrac{y \left(- 2 x - 3 y + 2\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x^{2}}{2} + x y - x + \tfrac{y^{2}}{2} - y + \tfrac{1}{2}\\\displaystyle \tfrac{x^{2}}{2} + x y - x + \tfrac{y^{2}}{2} - y + \tfrac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 3 x - 2 y + 2\right)}{2}\\\displaystyle \tfrac{x \left(- 3 x - 2 y + 2\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 x^{2} + 5 x y - x - 4 y^{2} + y\\\displaystyle y \left(- 4 x - y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 7 x^{2} - 13 x y + 11 x - 4 y^{2} + 7 y - 3\\\displaystyle 6 x^{2} + 11 x y - 9 x + 5 y^{2} - 8 y + 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2}\\\displaystyle x \left(- 2 x - 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 x^{2} + 5 x y - x - 4 y^{2} + y\\\displaystyle y \left(- 4 x - y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 7 x^{2} - 13 x y + 11 x - 4 y^{2} + 7 y - 3\\\displaystyle 6 x^{2} + 11 x y - 9 x + 5 y^{2} - 8 y + 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2}\\\displaystyle x \left(- 2 x - 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\nabla a\cdot\boldsymbol{v}(1,0)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(- 2 x + y\right)\\\displaystyle \tfrac{y \left(2 x - y\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2} + 4 x y - 4 x + y^{2} - 2 y + 1\\\displaystyle - \tfrac{3 x^{2}}{2} - 2 x y + 2 x - \tfrac{y^{2}}{2} + y - \tfrac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x^{2}\\\displaystyle \tfrac{x^{2}}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(- 2 x + y\right)\\\displaystyle \tfrac{y \left(2 x - y\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x^{2} + 4 x y - 4 x + y^{2} - 2 y + 1\\\displaystyle - \tfrac{3 x^{2}}{2} - 2 x y + 2 x - \tfrac{y^{2}}{2} + y - \tfrac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x^{2}\\\displaystyle \tfrac{x^{2}}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(- 3 x - 2 y + 1\right)\\\displaystyle 3 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 5 x^{2} + 11 x y - 8 x + 6 y^{2} - 9 y + 3\\\displaystyle - 4 x^{2} - 13 x y + 7 x - 7 y^{2} + 11 y - 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x \left(- x - 4 y + 1\right)\\\displaystyle - 4 x^{2} + 5 x y + x + 2 y^{2} - y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle y \left(- 3 x - 2 y + 1\right)\\\displaystyle 3 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 5 x^{2} + 11 x y - 8 x + 6 y^{2} - 9 y + 3\\\displaystyle - 4 x^{2} - 13 x y + 7 x - 7 y^{2} + 11 y - 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x \left(- x - 4 y + 1\right)\\\displaystyle - 4 x^{2} + 5 x y + x + 2 y^{2} - y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\nabla a\cdot\boldsymbol{v}(0,1)\)
\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y^{2}}{2}\\\displaystyle - y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x^{2}}{2} - 2 x y + x - \tfrac{3 y^{2}}{2} + 2 y - \tfrac{1}{2}\\\displaystyle x^{2} + 4 x y - 2 x + 3 y^{2} - 4 y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x + 2 y\right)}{2}\\\displaystyle x \left(x - 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{y^{2}}{2}\\\displaystyle - y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x^{2}}{2} - 2 x y + x - \tfrac{3 y^{2}}{2} + 2 y - \tfrac{1}{2}\\\displaystyle x^{2} + 4 x y - 2 x + 3 y^{2} - 4 y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- x + 2 y\right)}{2}\\\displaystyle x \left(x - 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{9} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(2 x - y\right)\\\displaystyle 4 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} + 12 x y - 4 x + 6 y^{2} - 8 y + 2\\\displaystyle - 4 x^{2} - 16 x y + 8 x - 12 y^{2} + 16 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 2 x^{2}\\\displaystyle 4 x \left(- x + 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{9} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(2 x - y\right)\\\displaystyle 4 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} + 12 x y - 4 x + 6 y^{2} - 8 y + 2\\\displaystyle - 4 x^{2} - 16 x y + 8 x - 12 y^{2} + 16 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 2 x^{2}\\\displaystyle 4 x \left(- x + 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{10} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 y \left(2 x - y\right)\\\displaystyle 2 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 12 x^{2} - 16 x y + 16 x - 4 y^{2} + 8 y - 4\\\displaystyle 6 x^{2} + 12 x y - 8 x + 2 y^{2} - 4 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 4 x^{2}\\\displaystyle 2 x \left(- x + 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{10} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 y \left(2 x - y\right)\\\displaystyle 2 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 12 x^{2} - 16 x y + 16 x - 4 y^{2} + 8 y - 4\\\displaystyle 6 x^{2} + 12 x y - 8 x + 2 y^{2} - 4 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 4 x^{2}\\\displaystyle 2 x \left(- x + 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{11} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(- 2 x - y + 2\right)\\\displaystyle - 4 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x^{2} - 12 x y + 4 x - 10 y^{2} + 12 y - 2\\\displaystyle 4 x^{2} + 16 x y - 8 x + 12 y^{2} - 16 y + 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x^{2} - 4 y^{2} + 4 y\\\displaystyle 4 x \left(x - 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{11} = \begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(- 2 x - y + 2\right)\\\displaystyle - 4 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x^{2} - 12 x y + 4 x - 10 y^{2} + 12 y - 2\\\displaystyle 4 x^{2} + 16 x y - 8 x + 12 y^{2} - 16 y + 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x^{2} - 4 y^{2} + 4 y\\\displaystyle 4 x \left(x - 2 y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{12} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 y \left(x + 2 y - 1\right)\\\displaystyle - 2 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 4 x^{2} - 12 x y + 8 x - 8 y^{2} + 12 y - 4\\\displaystyle 2 x^{2} + 8 x y - 4 x + 6 y^{2} - 8 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 4 x \left(x + 2 y - 1\right)\\\displaystyle 10 x^{2} - 8 x y - 4 x - 4 y^{2} + 4 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{12} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 y \left(x + 2 y - 1\right)\\\displaystyle - 2 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 4 x^{2} - 12 x y + 8 x - 8 y^{2} + 12 y - 4\\\displaystyle 2 x^{2} + 8 x y - 4 x + 6 y^{2} - 8 y + 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 4 x \left(x + 2 y - 1\right)\\\displaystyle 10 x^{2} - 8 x y - 4 x - 4 y^{2} + 4 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{13} = \begin{cases} \left(\begin{array}{c}\displaystyle - 4 x^{2} - 8 x y + 4 x + 10 y^{2} - 4 y\\\displaystyle 4 y \left(2 x + y - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 6 x^{2} + 8 x y - 8 x + 2 y^{2} - 4 y + 2\\\displaystyle - 8 x^{2} - 12 x y + 12 x - 4 y^{2} + 8 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x^{2}\\\displaystyle 4 x \left(2 x + y - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{13} = \begin{cases} \left(\begin{array}{c}\displaystyle - 4 x^{2} - 8 x y + 4 x + 10 y^{2} - 4 y\\\displaystyle 4 y \left(2 x + y - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 6 x^{2} + 8 x y - 8 x + 2 y^{2} - 4 y + 2\\\displaystyle - 8 x^{2} - 12 x y + 12 x - 4 y^{2} + 8 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x^{2}\\\displaystyle 4 x \left(2 x + y - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{14} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 y \left(- 2 x + y\right)\\\displaystyle - 4 x^{2} + 4 x - 2 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 12 x^{2} + 16 x y - 16 x + 4 y^{2} - 8 y + 4\\\displaystyle - 10 x^{2} - 12 x y + 12 x - 2 y^{2} + 4 y - 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 4 x^{2}\\\displaystyle 2 x \left(- x - 2 y + 2\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{14} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 y \left(- 2 x + y\right)\\\displaystyle - 4 x^{2} + 4 x - 2 y^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 12 x^{2} + 16 x y - 16 x + 4 y^{2} - 8 y + 4\\\displaystyle - 10 x^{2} - 12 x y + 12 x - 2 y^{2} + 4 y - 2\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 4 x^{2}\\\displaystyle 2 x \left(- x - 2 y + 2\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.