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Degree 2 Gopalakrishnan–Lederer–Schöberl 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: \(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{2}\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle x^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x y\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle x y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y^{2}\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle y^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{23}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(2 s_{0}^{2} - 3 s_{0} + 1)\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{399 x^{2}}{58} - \frac{90 x y}{29} - \frac{118 x}{29} + \frac{51 y^{2}}{58} - \frac{2 y}{29} + \frac{12}{29}&\displaystyle \frac{y \left(- 45 x + 17 y - 1\right)}{29}\\\displaystyle \frac{x \left(- 133 x + 45 y + 59\right)}{29}&\displaystyle - \frac{399 x^{2}}{58} + \frac{90 x y}{29} + \frac{118 x}{29} - \frac{51 y^{2}}{58} + \frac{2 y}{29} - \frac{12}{29}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{399 x^{2}}{58} - \frac{90 x y}{29} - \frac{118 x}{29} + \frac{51 y^{2}}{58} - \frac{2 y}{29} + \frac{12}{29}&\displaystyle \frac{y \left(- 45 x + 17 y - 1\right)}{29}\\\displaystyle \frac{x \left(- 133 x + 45 y + 59\right)}{29}&\displaystyle - \frac{399 x^{2}}{58} + \frac{90 x y}{29} + \frac{118 x}{29} - \frac{51 y^{2}}{58} + \frac{2 y}{29} - \frac{12}{29}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0} \left(2 s_{0} - 1\right))\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle \frac{51 x^{2}}{58} - \frac{90 x y}{29} - \frac{2 x}{29} + \frac{399 y^{2}}{58} - \frac{118 y}{29} + \frac{12}{29}&\displaystyle \frac{y \left(- 45 x + 133 y - 59\right)}{29}\\\displaystyle \frac{x \left(- 17 x + 45 y + 1\right)}{29}&\displaystyle - \frac{51 x^{2}}{58} + \frac{90 x y}{29} + \frac{2 x}{29} - \frac{399 y^{2}}{58} + \frac{118 y}{29} - \frac{12}{29}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle \frac{51 x^{2}}{58} - \frac{90 x y}{29} - \frac{2 x}{29} + \frac{399 y^{2}}{58} - \frac{118 y}{29} + \frac{12}{29}&\displaystyle \frac{y \left(- 45 x + 133 y - 59\right)}{29}\\\displaystyle \frac{x \left(- 17 x + 45 y + 1\right)}{29}&\displaystyle - \frac{51 x^{2}}{58} + \frac{90 x y}{29} + \frac{2 x}{29} - \frac{399 y^{2}}{58} + \frac{118 y}{29} - \frac{12}{29}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(4 s_{0} \left(1 - s_{0}\right))\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2}}{116} + \frac{165 x y}{29} - \frac{35 x}{29} + \frac{45 y^{2}}{116} - \frac{35 y}{29} + \frac{7}{29}&\displaystyle \frac{5 y \left(33 x + 3 y - 7\right)}{58}\\\displaystyle \frac{5 x \left(- 3 x - 33 y + 7\right)}{58}&\displaystyle - \frac{45 x^{2}}{116} - \frac{165 x y}{29} + \frac{35 x}{29} - \frac{45 y^{2}}{116} + \frac{35 y}{29} - \frac{7}{29}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2}}{116} + \frac{165 x y}{29} - \frac{35 x}{29} + \frac{45 y^{2}}{116} - \frac{35 y}{29} + \frac{7}{29}&\displaystyle \frac{5 y \left(33 x + 3 y - 7\right)}{58}\\\displaystyle \frac{5 x \left(- 3 x - 33 y + 7\right)}{58}&\displaystyle - \frac{45 x^{2}}{116} - \frac{165 x y}{29} + \frac{35 x}{29} - \frac{45 y^{2}}{116} + \frac{35 y}{29} - \frac{7}{29}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(2 s_{0}^{2} - 3 s_{0} + 1)\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle - \frac{223 x^{2}}{116} - \frac{141 x y}{29} + \frac{182 x}{87} - \frac{455 y^{2}}{116} + \frac{298 y}{87} - \frac{16}{29}&\displaystyle \frac{y \left(- 423 x - 455 y + 298\right)}{174}\\\displaystyle - \frac{3083 x^{2}}{174} - \frac{2643 x y}{58} + \frac{2258 x}{87} - 30 y^{2} + 36 y - 9&\displaystyle \frac{223 x^{2}}{116} + \frac{141 x y}{29} - \frac{182 x}{87} + \frac{455 y^{2}}{116} - \frac{298 y}{87} + \frac{16}{29}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle - \frac{223 x^{2}}{116} - \frac{141 x y}{29} + \frac{182 x}{87} - \frac{455 y^{2}}{116} + \frac{298 y}{87} - \frac{16}{29}&\displaystyle \frac{y \left(- 423 x - 455 y + 298\right)}{174}\\\displaystyle - \frac{3083 x^{2}}{174} - \frac{2643 x y}{58} + \frac{2258 x}{87} - 30 y^{2} + 36 y - 9&\displaystyle \frac{223 x^{2}}{116} + \frac{141 x y}{29} - \frac{182 x}{87} + \frac{455 y^{2}}{116} - \frac{298 y}{87} + \frac{16}{29}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0} \left(2 s_{0} - 1\right))\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle - \frac{37 x^{2}}{116} - \frac{39 x y}{29} + \frac{38 x}{87} - \frac{965 y^{2}}{116} + \frac{502 y}{87} - \frac{18}{29}&\displaystyle \frac{y \left(- 117 x - 965 y + 502\right)}{174}\\\displaystyle - \frac{137 x^{2}}{174} - \frac{657 x y}{58} + \frac{242 x}{87} - 30 y^{2} + 24 y - 3&\displaystyle \frac{37 x^{2}}{116} + \frac{39 x y}{29} - \frac{38 x}{87} + \frac{965 y^{2}}{116} - \frac{502 y}{87} + \frac{18}{29}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle - \frac{37 x^{2}}{116} - \frac{39 x y}{29} + \frac{38 x}{87} - \frac{965 y^{2}}{116} + \frac{502 y}{87} - \frac{18}{29}&\displaystyle \frac{y \left(- 117 x - 965 y + 502\right)}{174}\\\displaystyle - \frac{137 x^{2}}{174} - \frac{657 x y}{58} + \frac{242 x}{87} - 30 y^{2} + 24 y - 3&\displaystyle \frac{37 x^{2}}{116} + \frac{39 x y}{29} - \frac{38 x}{87} + \frac{965 y^{2}}{116} - \frac{502 y}{87} + \frac{18}{29}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(4 s_{0} \left(1 - s_{0}\right))\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle - \frac{95 x^{2}}{232} + \frac{135 x y}{58} - \frac{10 x}{87} + \frac{485 y^{2}}{232} - \frac{155 y}{87} + \frac{11}{58}&\displaystyle \frac{5 y \left(81 x + 97 y - 62\right)}{348}\\\displaystyle - \frac{775 x^{2}}{348} + \frac{1605 x y}{116} + \frac{5 x}{87} + 15 y^{2} - 15 y + \frac{3}{2}&\displaystyle \frac{95 x^{2}}{232} - \frac{135 x y}{58} + \frac{10 x}{87} - \frac{485 y^{2}}{232} + \frac{155 y}{87} - \frac{11}{58}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle - \frac{95 x^{2}}{232} + \frac{135 x y}{58} - \frac{10 x}{87} + \frac{485 y^{2}}{232} - \frac{155 y}{87} + \frac{11}{58}&\displaystyle \frac{5 y \left(81 x + 97 y - 62\right)}{348}\\\displaystyle - \frac{775 x^{2}}{348} + \frac{1605 x y}{116} + \frac{5 x}{87} + 15 y^{2} - 15 y + \frac{3}{2}&\displaystyle \frac{95 x^{2}}{232} - \frac{135 x y}{58} + \frac{10 x}{87} - \frac{485 y^{2}}{232} + \frac{155 y}{87} - \frac{11}{58}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(2 s_{0}^{2} - 3 s_{0} + 1)\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle - \frac{455 x^{2}}{116} - \frac{141 x y}{29} + \frac{298 x}{87} - \frac{223 y^{2}}{116} + \frac{182 y}{87} - \frac{16}{29}&\displaystyle 30 x^{2} + \frac{2643 x y}{58} - 36 x + \frac{3083 y^{2}}{174} - \frac{2258 y}{87} + 9\\\displaystyle \frac{x \left(455 x + 423 y - 298\right)}{174}&\displaystyle \frac{455 x^{2}}{116} + \frac{141 x y}{29} - \frac{298 x}{87} + \frac{223 y^{2}}{116} - \frac{182 y}{87} + \frac{16}{29}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle - \frac{455 x^{2}}{116} - \frac{141 x y}{29} + \frac{298 x}{87} - \frac{223 y^{2}}{116} + \frac{182 y}{87} - \frac{16}{29}&\displaystyle 30 x^{2} + \frac{2643 x y}{58} - 36 x + \frac{3083 y^{2}}{174} - \frac{2258 y}{87} + 9\\\displaystyle \frac{x \left(455 x + 423 y - 298\right)}{174}&\displaystyle \frac{455 x^{2}}{116} + \frac{141 x y}{29} - \frac{298 x}{87} + \frac{223 y^{2}}{116} - \frac{182 y}{87} + \frac{16}{29}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0} \left(2 s_{0} - 1\right))\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle - \frac{965 x^{2}}{116} - \frac{39 x y}{29} + \frac{502 x}{87} - \frac{37 y^{2}}{116} + \frac{38 y}{87} - \frac{18}{29}&\displaystyle 30 x^{2} + \frac{657 x y}{58} - 24 x + \frac{137 y^{2}}{174} - \frac{242 y}{87} + 3\\\displaystyle \frac{x \left(965 x + 117 y - 502\right)}{174}&\displaystyle \frac{965 x^{2}}{116} + \frac{39 x y}{29} - \frac{502 x}{87} + \frac{37 y^{2}}{116} - \frac{38 y}{87} + \frac{18}{29}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle - \frac{965 x^{2}}{116} - \frac{39 x y}{29} + \frac{502 x}{87} - \frac{37 y^{2}}{116} + \frac{38 y}{87} - \frac{18}{29}&\displaystyle 30 x^{2} + \frac{657 x y}{58} - 24 x + \frac{137 y^{2}}{174} - \frac{242 y}{87} + 3\\\displaystyle \frac{x \left(965 x + 117 y - 502\right)}{174}&\displaystyle \frac{965 x^{2}}{116} + \frac{39 x y}{29} - \frac{502 x}{87} + \frac{37 y^{2}}{116} - \frac{38 y}{87} + \frac{18}{29}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(4 s_{0} \left(1 - s_{0}\right))\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle \frac{485 x^{2}}{232} + \frac{135 x y}{58} - \frac{155 x}{87} - \frac{95 y^{2}}{232} - \frac{10 y}{87} + \frac{11}{58}&\displaystyle - 15 x^{2} - \frac{1605 x y}{116} + 15 x + \frac{775 y^{2}}{348} - \frac{5 y}{87} - \frac{3}{2}\\\displaystyle \frac{5 x \left(- 97 x - 81 y + 62\right)}{348}&\displaystyle - \frac{485 x^{2}}{232} - \frac{135 x y}{58} + \frac{155 x}{87} + \frac{95 y^{2}}{232} + \frac{10 y}{87} - \frac{11}{58}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle \frac{485 x^{2}}{232} + \frac{135 x y}{58} - \frac{155 x}{87} - \frac{95 y^{2}}{232} - \frac{10 y}{87} + \frac{11}{58}&\displaystyle - 15 x^{2} - \frac{1605 x y}{116} + 15 x + \frac{775 y^{2}}{348} - \frac{5 y}{87} - \frac{3}{2}\\\displaystyle \frac{5 x \left(- 97 x - 81 y + 62\right)}{348}&\displaystyle - \frac{485 x^{2}}{232} - \frac{135 x y}{58} + \frac{155 x}{87} + \frac{95 y^{2}}{232} + \frac{10 y}{87} - \frac{11}{58}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36&\displaystyle 0\\\displaystyle 0&\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36&\displaystyle 0\\\displaystyle 0&\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(s_{0} \left(2 s_{0} - 1\right))\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 90 x^{2} - 60 x + 6&\displaystyle 0\\\displaystyle 0&\displaystyle 90 x^{2} - 60 x + 6\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 90 x^{2} - 60 x + 6&\displaystyle 0\\\displaystyle 0&\displaystyle 90 x^{2} - 60 x + 6\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(s_{1} \left(2 s_{1} - 1\right))\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 90 y^{2} - 60 y + 6&\displaystyle 0\\\displaystyle 0&\displaystyle 90 y^{2} - 60 y + 6\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 90 y^{2} - 60 y + 6&\displaystyle 0\\\displaystyle 0&\displaystyle 90 y^{2} - 60 y + 6\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(4 s_{0} s_{1})\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2}}{2} + 90 x y - 30 x + \frac{45 y^{2}}{2} - 30 y + 6&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{45 x^{2}}{2} + 90 x y - 30 x + \frac{45 y^{2}}{2} - 30 y + 6\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2}}{2} + 90 x y - 30 x + \frac{45 y^{2}}{2} - 30 y + 6&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{45 x^{2}}{2} + 90 x y - 30 x + \frac{45 y^{2}}{2} - 30 y + 6\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2}}{2} - 45 x y - 15 x - 45 y^{2} + 45 y - \frac{3}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{45 x^{2}}{2} - 45 x y - 15 x - 45 y^{2} + 45 y - \frac{3}{2}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2}}{2} - 45 x y - 15 x - 45 y^{2} + 45 y - \frac{3}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{45 x^{2}}{2} - 45 x y - 15 x - 45 y^{2} + 45 y - \frac{3}{2}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle - 45 x^{2} - 45 x y + 45 x + \frac{45 y^{2}}{2} - 15 y - \frac{3}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle - 45 x^{2} - 45 x y + 45 x + \frac{45 y^{2}}{2} - 15 y - \frac{3}{2}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle - 45 x^{2} - 45 x y + 45 x + \frac{45 y^{2}}{2} - 15 y - \frac{3}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle - 45 x^{2} - 45 x y + 45 x + \frac{45 y^{2}}{2} - 15 y - \frac{3}{2}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle - \frac{s_{0}^{2}}{2} - s_{0} s_{1} + s_{0} - \frac{s_{1}^{2}}{2} + s_{1} - \frac{1}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{s_{0}^{2}}{2} + s_{0} s_{1} - s_{0} + \frac{s_{1}^{2}}{2} - s_{1} + \frac{1}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle - \frac{4140 x^{2}}{29} - \frac{8520 x y}{29} + \frac{5920 x}{29} - \frac{4140 y^{2}}{29} + \frac{5920 y}{29} - \frac{1880}{29}&\displaystyle \frac{40 y \left(24 x + 18 y - 13\right)}{29}\\\displaystyle \frac{40 x \left(- 18 x - 24 y + 13\right)}{29}&\displaystyle \frac{4140 x^{2}}{29} + \frac{8520 x y}{29} - \frac{5920 x}{29} + \frac{4140 y^{2}}{29} - \frac{5920 y}{29} + \frac{1880}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle - \frac{4140 x^{2}}{29} - \frac{8520 x y}{29} + \frac{5920 x}{29} - \frac{4140 y^{2}}{29} + \frac{5920 y}{29} - \frac{1880}{29}&\displaystyle \frac{40 y \left(24 x + 18 y - 13\right)}{29}\\\displaystyle \frac{40 x \left(- 18 x - 24 y + 13\right)}{29}&\displaystyle \frac{4140 x^{2}}{29} + \frac{8520 x y}{29} - \frac{5920 x}{29} + \frac{4140 y^{2}}{29} - \frac{5920 y}{29} + \frac{1880}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{16}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0} \left(- s_{0} - s_{1} + 1\right)}{2}&\displaystyle 0\\\displaystyle s_{0} \left(- s_{0} - s_{1} + 1\right)&\displaystyle \frac{s_{0} \left(s_{0} + s_{1} - 1\right)}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle - \frac{2220 x^{2}}{29} - \frac{2400 x y}{29} + \frac{1880 x}{29} - \frac{480 y^{2}}{29} + \frac{720 y}{29} - \frac{260}{29}&\displaystyle \frac{40 y \left(- 30 x - 8 y + 9\right)}{29}\\\displaystyle \frac{40 x \left(- 224 x - 231 y + 194\right)}{29}&\displaystyle \frac{2220 x^{2}}{29} + \frac{2400 x y}{29} - \frac{1880 x}{29} + \frac{480 y^{2}}{29} - \frac{720 y}{29} + \frac{260}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle - \frac{2220 x^{2}}{29} - \frac{2400 x y}{29} + \frac{1880 x}{29} - \frac{480 y^{2}}{29} + \frac{720 y}{29} - \frac{260}{29}&\displaystyle \frac{40 y \left(- 30 x - 8 y + 9\right)}{29}\\\displaystyle \frac{40 x \left(- 224 x - 231 y + 194\right)}{29}&\displaystyle \frac{2220 x^{2}}{29} + \frac{2400 x y}{29} - \frac{1880 x}{29} + \frac{480 y^{2}}{29} - \frac{720 y}{29} + \frac{260}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{1} \left(- s_{0} - s_{1} + 1\right)}{2}&\displaystyle s_{1} \left(s_{0} + s_{1} - 1\right)\\\displaystyle 0&\displaystyle \frac{s_{1} \left(s_{0} + s_{1} - 1\right)}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle - \frac{480 x^{2}}{29} - \frac{2400 x y}{29} + \frac{720 x}{29} - \frac{2220 y^{2}}{29} + \frac{1880 y}{29} - \frac{260}{29}&\displaystyle \frac{40 y \left(231 x + 224 y - 194\right)}{29}\\\displaystyle \frac{40 x \left(8 x + 30 y - 9\right)}{29}&\displaystyle \frac{480 x^{2}}{29} + \frac{2400 x y}{29} - \frac{720 x}{29} + \frac{2220 y^{2}}{29} - \frac{1880 y}{29} + \frac{260}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle - \frac{480 x^{2}}{29} - \frac{2400 x y}{29} + \frac{720 x}{29} - \frac{2220 y^{2}}{29} + \frac{1880 y}{29} - \frac{260}{29}&\displaystyle \frac{40 y \left(231 x + 224 y - 194\right)}{29}\\\displaystyle \frac{40 x \left(8 x + 30 y - 9\right)}{29}&\displaystyle \frac{480 x^{2}}{29} + \frac{2400 x y}{29} - \frac{720 x}{29} + \frac{2220 y^{2}}{29} - \frac{1880 y}{29} + \frac{260}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{18}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0} \left(s_{0} + s_{1} - 1\right)}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{s_{0} \left(- s_{0} - s_{1} + 1\right)}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{cc}\displaystyle \frac{11940 x^{2}}{29} + \frac{12720 x y}{29} - \frac{13560 x}{29} - \frac{240 y^{2}}{29} - \frac{1960 y}{29} + \frac{1900}{29}&\displaystyle \frac{40 y \left(- 102 x - 4 y + 19\right)}{29}\\\displaystyle \frac{40 x \left(- 112 x - 159 y + 126\right)}{29}&\displaystyle - \frac{11940 x^{2}}{29} - \frac{12720 x y}{29} + \frac{13560 x}{29} + \frac{240 y^{2}}{29} + \frac{1960 y}{29} - \frac{1900}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{cc}\displaystyle \frac{11940 x^{2}}{29} + \frac{12720 x y}{29} - \frac{13560 x}{29} - \frac{240 y^{2}}{29} - \frac{1960 y}{29} + \frac{1900}{29}&\displaystyle \frac{40 y \left(- 102 x - 4 y + 19\right)}{29}\\\displaystyle \frac{40 x \left(- 112 x - 159 y + 126\right)}{29}&\displaystyle - \frac{11940 x^{2}}{29} - \frac{12720 x y}{29} + \frac{13560 x}{29} + \frac{240 y^{2}}{29} + \frac{1960 y}{29} - \frac{1900}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{19}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0}^{2}}{2}&\displaystyle 0\\\displaystyle s_{0}^{2}&\displaystyle - \frac{s_{0}^{2}}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{cc}\displaystyle \frac{1590 x^{2}}{29} + \frac{120 x y}{29} - \frac{1080 x}{29} - \frac{150 y^{2}}{29} + \frac{80 y}{29} + \frac{100}{29}&\displaystyle \frac{20 y \left(3 x - 5 y + 2\right)}{29}\\\displaystyle \frac{20 x \left(121 x - 3 y - 60\right)}{29}&\displaystyle - \frac{1590 x^{2}}{29} - \frac{120 x y}{29} + \frac{1080 x}{29} + \frac{150 y^{2}}{29} - \frac{80 y}{29} - \frac{100}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{cc}\displaystyle \frac{1590 x^{2}}{29} + \frac{120 x y}{29} - \frac{1080 x}{29} - \frac{150 y^{2}}{29} + \frac{80 y}{29} + \frac{100}{29}&\displaystyle \frac{20 y \left(3 x - 5 y + 2\right)}{29}\\\displaystyle \frac{20 x \left(121 x - 3 y - 60\right)}{29}&\displaystyle - \frac{1590 x^{2}}{29} - \frac{120 x y}{29} + \frac{1080 x}{29} + \frac{150 y^{2}}{29} - \frac{80 y}{29} - \frac{100}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{20}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0} s_{1}}{2}&\displaystyle - s_{0} s_{1}\\\displaystyle 0&\displaystyle - \frac{s_{0} s_{1}}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{cc}\displaystyle \frac{1410 x^{2}}{29} + \frac{4440 x y}{29} - \frac{1680 x}{29} - \frac{330 y^{2}}{29} - \frac{520 y}{29} + \frac{220}{29}&\displaystyle \frac{20 y \left(- 411 x - 11 y + 74\right)}{29}\\\displaystyle \frac{20 x \left(- 47 x - 111 y + 42\right)}{29}&\displaystyle - \frac{1410 x^{2}}{29} - \frac{4440 x y}{29} + \frac{1680 x}{29} + \frac{330 y^{2}}{29} + \frac{520 y}{29} - \frac{220}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{cc}\displaystyle \frac{1410 x^{2}}{29} + \frac{4440 x y}{29} - \frac{1680 x}{29} - \frac{330 y^{2}}{29} - \frac{520 y}{29} + \frac{220}{29}&\displaystyle \frac{20 y \left(- 411 x - 11 y + 74\right)}{29}\\\displaystyle \frac{20 x \left(- 47 x - 111 y + 42\right)}{29}&\displaystyle - \frac{1410 x^{2}}{29} - \frac{4440 x y}{29} + \frac{1680 x}{29} + \frac{330 y^{2}}{29} + \frac{520 y}{29} - \frac{220}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{1} \left(s_{0} + s_{1} - 1\right)}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle \frac{s_{1} \left(- s_{0} - s_{1} + 1\right)}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{cc}\displaystyle - \frac{240 x^{2}}{29} + \frac{12720 x y}{29} - \frac{1960 x}{29} + \frac{11940 y^{2}}{29} - \frac{13560 y}{29} + \frac{1900}{29}&\displaystyle \frac{40 y \left(159 x + 112 y - 126\right)}{29}\\\displaystyle \frac{40 x \left(4 x + 102 y - 19\right)}{29}&\displaystyle \frac{240 x^{2}}{29} - \frac{12720 x y}{29} + \frac{1960 x}{29} - \frac{11940 y^{2}}{29} + \frac{13560 y}{29} - \frac{1900}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{cc}\displaystyle - \frac{240 x^{2}}{29} + \frac{12720 x y}{29} - \frac{1960 x}{29} + \frac{11940 y^{2}}{29} - \frac{13560 y}{29} + \frac{1900}{29}&\displaystyle \frac{40 y \left(159 x + 112 y - 126\right)}{29}\\\displaystyle \frac{40 x \left(4 x + 102 y - 19\right)}{29}&\displaystyle \frac{240 x^{2}}{29} - \frac{12720 x y}{29} + \frac{1960 x}{29} - \frac{11940 y^{2}}{29} + \frac{13560 y}{29} - \frac{1900}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{22}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0} s_{1}}{2}&\displaystyle 0\\\displaystyle s_{0} s_{1}&\displaystyle - \frac{s_{0} s_{1}}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{cc}\displaystyle - \frac{330 x^{2}}{29} + \frac{4440 x y}{29} - \frac{520 x}{29} + \frac{1410 y^{2}}{29} - \frac{1680 y}{29} + \frac{220}{29}&\displaystyle \frac{20 y \left(111 x + 47 y - 42\right)}{29}\\\displaystyle \frac{20 x \left(11 x + 411 y - 74\right)}{29}&\displaystyle \frac{330 x^{2}}{29} - \frac{4440 x y}{29} + \frac{520 x}{29} - \frac{1410 y^{2}}{29} + \frac{1680 y}{29} - \frac{220}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{cc}\displaystyle - \frac{330 x^{2}}{29} + \frac{4440 x y}{29} - \frac{520 x}{29} + \frac{1410 y^{2}}{29} - \frac{1680 y}{29} + \frac{220}{29}&\displaystyle \frac{20 y \left(111 x + 47 y - 42\right)}{29}\\\displaystyle \frac{20 x \left(11 x + 411 y - 74\right)}{29}&\displaystyle \frac{330 x^{2}}{29} - \frac{4440 x y}{29} + \frac{520 x}{29} - \frac{1410 y^{2}}{29} + \frac{1680 y}{29} - \frac{220}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{23}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{1}^{2}}{2}&\displaystyle - s_{1}^{2}\\\displaystyle 0&\displaystyle - \frac{s_{1}^{2}}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{cc}\displaystyle - \frac{150 x^{2}}{29} + \frac{120 x y}{29} + \frac{80 x}{29} + \frac{1590 y^{2}}{29} - \frac{1080 y}{29} + \frac{100}{29}&\displaystyle \frac{20 y \left(3 x - 121 y + 60\right)}{29}\\\displaystyle \frac{20 x \left(5 x - 3 y - 2\right)}{29}&\displaystyle \frac{150 x^{2}}{29} - \frac{120 x y}{29} - \frac{80 x}{29} - \frac{1590 y^{2}}{29} + \frac{1080 y}{29} - \frac{100}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{cc}\displaystyle - \frac{150 x^{2}}{29} + \frac{120 x y}{29} + \frac{80 x}{29} + \frac{1590 y^{2}}{29} - \frac{1080 y}{29} + \frac{100}{29}&\displaystyle \frac{20 y \left(3 x - 121 y + 60\right)}{29}\\\displaystyle \frac{20 x \left(5 x - 3 y - 2\right)}{29}&\displaystyle \frac{150 x^{2}}{29} - \frac{120 x y}{29} - \frac{80 x}{29} - \frac{1590 y^{2}}{29} + \frac{1080 y}{29} - \frac{100}{29}\end{array}\right)\)
This DOF is associated with face 0 of the reference element.