an encyclopedia of finite element definitions
Degree 3 Hsieh–Clough–Tocher on a triangle
◀ Back to Hsieh–Clough–Tocher definition page
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\begin{cases} 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} x&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\y&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} x^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x^{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} x y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x y&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\y^{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} x^{3}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x^{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{3}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} x^{2} y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x^{2} y&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{2} y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} x y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x y^{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} y^{3}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\y^{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\y^{3}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} - 23 x^{3} + 24 x^{2} y - 12 x y^{2} + 36 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 28 x^{3} + 12 x^{2} y + 9 x^{2} - 3 x + 32 y^{3} + 12 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 15 x^{2} - 33 x y^{2} + 30 x y + 22 y^{3} + 21 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} 22 x^{3} - 21 x^{2} y - 12 x y^{2} + 30 x y - 24 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\32 x^{3} + 12 x^{2} y - 21 x^{2} + 12 x - 28 y^{3} - 3 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\15 x^{2} + 12 x y^{2} - 23 y^{3} - 9 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
- \(\mathcal{L}=\{l_0,...,l_{11}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} - 3 x y^{2} + 3 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + 21 x^{2} y - 15 x^{2} + 21 x y^{2} - 30 x y + 9 x + 7 y^{3} - 15 y^{2} + 9 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x^{3} - 3 x^{2} y - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} - 3 x y^{2} + 3 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + 21 x^{2} y - 15 x^{2} + 21 x y^{2} - 30 x y + 9 x + 7 y^{3} - 15 y^{2} + 9 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x^{3} - 3 x^{2} y - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)\)
\(\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} - \tfrac{x y^{2}}{2} + x + \tfrac{7 y^{3}}{3} - \tfrac{3 y^{2}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{13 x^{3}}{6} + 4 x^{2} y - \tfrac{9 x^{2}}{2} + \tfrac{3 x y^{2}}{2} - 4 x y + \tfrac{5 x}{2} - \tfrac{y^{3}}{3} + \tfrac{y^{2}}{2} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(- 7 x^{2} + 12 x y - 3 x + 12 y^{2} - 18 y + 6\right)}{6}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} - \tfrac{x y^{2}}{2} + x + \tfrac{7 y^{3}}{3} - \tfrac{3 y^{2}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{13 x^{3}}{6} + 4 x^{2} y - \tfrac{9 x^{2}}{2} + \tfrac{3 x y^{2}}{2} - 4 x y + \tfrac{5 x}{2} - \tfrac{y^{3}}{3} + \tfrac{y^{2}}{2} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(- 7 x^{2} + 12 x y - 3 x + 12 y^{2} - 18 y + 6\right)}{6}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)\)
\(\displaystyle \phi_{2} = \begin{cases} \tfrac{y \left(12 x^{2} + 12 x y - 18 x - 7 y^{2} - 3 y + 6\right)}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{x^{3}}{3} + \tfrac{3 x^{2} y}{2} + \tfrac{x^{2}}{2} + 4 x y^{2} - 4 x y + \tfrac{13 y^{3}}{6} - \tfrac{9 y^{2}}{2} + \tfrac{5 y}{2} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{7 x^{3}}{3} - \tfrac{x^{2} y}{2} - \tfrac{3 x^{2}}{2} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{2} = \begin{cases} \tfrac{y \left(12 x^{2} + 12 x y - 18 x - 7 y^{2} - 3 y + 6\right)}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{x^{3}}{3} + \tfrac{3 x^{2} y}{2} + \tfrac{x^{2}}{2} + 4 x y^{2} - 4 x y + \tfrac{13 y^{3}}{6} - \tfrac{9 y^{2}}{2} + \tfrac{5 y}{2} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{7 x^{3}}{3} - \tfrac{x^{2} y}{2} - \tfrac{3 x^{2}}{2} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} + \tfrac{3 x y^{2}}{2} - \tfrac{y^{3}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{9 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 9 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{5 y^{3}}{2} + 6 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(- 5 x + 3 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} + \tfrac{3 x y^{2}}{2} - \tfrac{y^{3}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{9 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 9 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{5 y^{3}}{2} + 6 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(- 5 x + 3 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)\)
\(\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} - \tfrac{x y^{2}}{4} + \tfrac{y^{3}}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{17 x^{3}}{12} + \tfrac{7 x^{2} y}{4} - 2 x^{2} + \tfrac{7 x y^{2}}{4} - \tfrac{5 x y}{2} + \tfrac{3 x}{4} + \tfrac{5 y^{3}}{12} - y^{2} + \tfrac{3 y}{4} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \cdot \left(13 x - 3 y - 12\right)}{12}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} - \tfrac{x y^{2}}{4} + \tfrac{y^{3}}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{17 x^{3}}{12} + \tfrac{7 x^{2} y}{4} - 2 x^{2} + \tfrac{7 x y^{2}}{4} - \tfrac{5 x y}{2} + \tfrac{3 x}{4} + \tfrac{5 y^{3}}{12} - y^{2} + \tfrac{3 y}{4} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \cdot \left(13 x - 3 y - 12\right)}{12}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)\)
\(\displaystyle \phi_{5} = \begin{cases} \tfrac{y \left(24 x^{2} + 21 x y - 12 x - 13 y^{2} + 6 y\right)}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{23 x^{3}}{12} - \tfrac{23 x^{2} y}{4} + \tfrac{9 x^{2}}{2} - \tfrac{25 x y^{2}}{4} + \tfrac{19 x y}{2} - \tfrac{13 x}{4} - \tfrac{17 y^{3}}{12} + \tfrac{7 y^{2}}{2} - \tfrac{11 y}{4} + \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \cdot \left(5 x + 27 y - 6\right)}{12}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{5} = \begin{cases} \tfrac{y \left(24 x^{2} + 21 x y - 12 x - 13 y^{2} + 6 y\right)}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{23 x^{3}}{12} - \tfrac{23 x^{2} y}{4} + \tfrac{9 x^{2}}{2} - \tfrac{25 x y^{2}}{4} + \tfrac{19 x y}{2} - \tfrac{13 x}{4} - \tfrac{17 y^{3}}{12} + \tfrac{7 y^{2}}{2} - \tfrac{11 y}{4} + \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \cdot \left(5 x + 27 y - 6\right)}{12}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{6} = \begin{cases} \tfrac{y^{2} \cdot \left(3 x - 5 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{5 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 6 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{9 y^{3}}{2} + 9 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{x^{3}}{2} + \tfrac{3 x^{2} y}{2} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{6} = \begin{cases} \tfrac{y^{2} \cdot \left(3 x - 5 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{5 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 6 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{9 y^{3}}{2} + 9 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{x^{3}}{2} + \tfrac{3 x^{2} y}{2} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)
\(\displaystyle \phi_{7} = \begin{cases} \tfrac{y^{2} \cdot \left(27 x + 5 y - 6\right)}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{17 x^{3}}{12} - \tfrac{25 x^{2} y}{4} + \tfrac{7 x^{2}}{2} - \tfrac{23 x y^{2}}{4} + \tfrac{19 x y}{2} - \tfrac{11 x}{4} - \tfrac{23 y^{3}}{12} + \tfrac{9 y^{2}}{2} - \tfrac{13 y}{4} + \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(- 13 x^{2} + 21 x y + 6 x + 24 y^{2} - 12 y\right)}{12}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{7} = \begin{cases} \tfrac{y^{2} \cdot \left(27 x + 5 y - 6\right)}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{17 x^{3}}{12} - \tfrac{25 x^{2} y}{4} + \tfrac{7 x^{2}}{2} - \tfrac{23 x y^{2}}{4} + \tfrac{19 x y}{2} - \tfrac{11 x}{4} - \tfrac{23 y^{3}}{12} + \tfrac{9 y^{2}}{2} - \tfrac{13 y}{4} + \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(- 13 x^{2} + 21 x y + 6 x + 24 y^{2} - 12 y\right)}{12}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)
\(\displaystyle \phi_{8} = \begin{cases} \tfrac{y^{2} \left(- 3 x + 13 y - 12\right)}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{5 x^{3}}{12} + \tfrac{7 x^{2} y}{4} - x^{2} + \tfrac{7 x y^{2}}{4} - \tfrac{5 x y}{2} + \tfrac{3 x}{4} + \tfrac{17 y^{3}}{12} - 2 y^{2} + \tfrac{3 y}{4} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{3}}{12} - \tfrac{x^{2} y}{4} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{8} = \begin{cases} \tfrac{y^{2} \left(- 3 x + 13 y - 12\right)}{12}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{5 x^{3}}{12} + \tfrac{7 x^{2} y}{4} - x^{2} + \tfrac{7 x y^{2}}{4} - \tfrac{5 x y}{2} + \tfrac{3 x}{4} + \tfrac{17 y^{3}}{12} - 2 y^{2} + \tfrac{3 y}{4} - \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{3}}{12} - \tfrac{x^{2} y}{4} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
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.
\(\displaystyle \phi_{9} = \begin{cases} \sqrt{2} y^{2} \left(x - \tfrac{y}{3}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{\sqrt{2} \left(- 5 x^{3} - 21 x^{2} y + 12 x^{2} - 21 x y^{2} + 30 x y - 9 x - 5 y^{3} + 12 y^{2} - 9 y + 2\right)}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\sqrt{2} x^{2} \left(- \tfrac{x}{3} + y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
where \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \phi_{9} = \begin{cases} \sqrt{2} y^{2} \left(x - \tfrac{y}{3}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{\sqrt{2} \left(- 5 x^{3} - 21 x^{2} y + 12 x^{2} - 21 x y^{2} + 30 x y - 9 x - 5 y^{3} + 12 y^{2} - 9 y + 2\right)}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\sqrt{2} x^{2} \left(- \tfrac{x}{3} + y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{10}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}\)
where \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \phi_{10} = \begin{cases} y^{2} \cdot \left(2 x + \tfrac{8 y}{3} - 2\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{2 x^{3}}{3} - 4 x^{2} y + 2 x^{2} - 6 x y^{2} + 8 x y - 2 x - \tfrac{8 y^{3}}{3} + 6 y^{2} - 4 y + \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{2 x \left(- 5 x^{2} + 6 x y + 3 x + 6 y^{2} - 6 y\right)}{3}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 1 of the reference element.
where \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \phi_{10} = \begin{cases} y^{2} \cdot \left(2 x + \tfrac{8 y}{3} - 2\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{2 x^{3}}{3} - 4 x^{2} y + 2 x^{2} - 6 x y^{2} + 8 x y - 2 x - \tfrac{8 y^{3}}{3} + 6 y^{2} - 4 y + \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{2 x \left(- 5 x^{2} + 6 x y + 3 x + 6 y^{2} - 6 y\right)}{3}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
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.
\(\displaystyle \phi_{11} = \begin{cases} \tfrac{2 y \left(- 6 x^{2} - 6 x y + 6 x + 5 y^{2} - 3 y\right)}{3}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{8 x^{3}}{3} + 6 x^{2} y - 6 x^{2} + 4 x y^{2} - 8 x y + 4 x + \tfrac{2 y^{3}}{3} - 2 y^{2} + 2 y - \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{2} \left(- \tfrac{8 x}{3} - 2 y + 2\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.
where \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \phi_{11} = \begin{cases} \tfrac{2 y \left(- 6 x^{2} - 6 x y + 6 x + 5 y^{2} - 3 y\right)}{3}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{8 x^{3}}{3} + 6 x^{2} y - 6 x^{2} + 4 x y^{2} - 8 x y + 4 x + \tfrac{2 y^{3}}{3} - 2 y^{2} + 2 y - \tfrac{2}{3}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{2} \left(- \tfrac{8 x}{3} - 2 y + 2\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with edge 2 of the reference element.