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Degree 3 Arnold–Winther 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: \(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{2}\\\displaystyle x^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x y\\\displaystyle x y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y^{2}\\\displaystyle y^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 20 y^{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 12 x y^{2}&\displaystyle - 4 y^{3}\\\displaystyle - 4 y^{3}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 6 x^{2} y&\displaystyle - 6 x y^{2}\\\displaystyle - 6 x y^{2}&\displaystyle 2 y^{3}\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 2 x^{3}&\displaystyle - 6 x^{2} y\\\displaystyle - 6 x^{2} y&\displaystyle 6 x y^{2}\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{3}\\\displaystyle x^{3}&\displaystyle - 3 x^{2} y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{3}\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{23}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(0,0)\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 2 x^{3} + 9 x^{2} y - 3 x^{2} - 10 y^{3} + 18 y^{2} - 9 y + 1&\displaystyle 3 x y \left(- 2 x - 3 y + 2\right)\\\displaystyle 3 x y \left(- 2 x - 3 y + 2\right)&\displaystyle 3 y^{2} \left(2 x + y - 1\right)\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 2 x^{3} + 9 x^{2} y - 3 x^{2} - 10 y^{3} + 18 y^{2} - 9 y + 1&\displaystyle 3 x y \left(- 2 x - 3 y + 2\right)\\\displaystyle 3 x y \left(- 2 x - 3 y + 2\right)&\displaystyle 3 y^{2} \left(2 x + y - 1\right)\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(0,0)\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 3 x \left(5 x^{2} + 15 x y - 8 x + 10 y^{2} - 12 y + 3\right)&\displaystyle - 10 x^{3} - 45 x^{2} y + 18 x^{2} - 45 x y^{2} + 48 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1\\\displaystyle - 10 x^{3} - 45 x^{2} y + 18 x^{2} - 45 x y^{2} + 48 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1&\displaystyle 3 y \left(10 x^{2} + 15 x y - 12 x + 5 y^{2} - 8 y + 3\right)\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 3 x \left(5 x^{2} + 15 x y - 8 x + 10 y^{2} - 12 y + 3\right)&\displaystyle - 10 x^{3} - 45 x^{2} y + 18 x^{2} - 45 x y^{2} + 48 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1\\\displaystyle - 10 x^{3} - 45 x^{2} y + 18 x^{2} - 45 x y^{2} + 48 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1&\displaystyle 3 y \left(10 x^{2} + 15 x y - 12 x + 5 y^{2} - 8 y + 3\right)\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,0)\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle 3 x^{2} \left(x + 2 y - 1\right)&\displaystyle 3 x y \left(- 3 x - 2 y + 2\right)\\\displaystyle 3 x y \left(- 3 x - 2 y + 2\right)&\displaystyle - 10 x^{3} + 18 x^{2} + 9 x y^{2} - 9 x + 2 y^{3} - 3 y^{2} + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle 3 x^{2} \left(x + 2 y - 1\right)&\displaystyle 3 x y \left(- 3 x - 2 y + 2\right)\\\displaystyle 3 x y \left(- 3 x - 2 y + 2\right)&\displaystyle - 10 x^{3} + 18 x^{2} + 9 x y^{2} - 9 x + 2 y^{3} - 3 y^{2} + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(1,0)\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle x^{2} \left(- 2 x - 9 y + 3\right)&\displaystyle 3 x y \left(2 x + 3 y - 2\right)\\\displaystyle 3 x y \left(2 x + 3 y - 2\right)&\displaystyle 3 y^{2} \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle x^{2} \left(- 2 x - 9 y + 3\right)&\displaystyle 3 x y \left(2 x + 3 y - 2\right)\\\displaystyle 3 x y \left(2 x + 3 y - 2\right)&\displaystyle 3 y^{2} \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(1,0)\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 9 x^{2} \left(- x - 2 y + 1\right)&\displaystyle x \left(10 x^{2} + 27 x y - 12 x + 18 y^{2} - 18 y + 3\right)\\\displaystyle x \left(10 x^{2} + 27 x y - 12 x + 18 y^{2} - 18 y + 3\right)&\displaystyle 3 y \left(- 10 x^{2} - 9 x y + 8 x - 2 y^{2} + 3 y - 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 9 x^{2} \left(- x - 2 y + 1\right)&\displaystyle x \left(10 x^{2} + 27 x y - 12 x + 18 y^{2} - 18 y + 3\right)\\\displaystyle x \left(10 x^{2} + 27 x y - 12 x + 18 y^{2} - 18 y + 3\right)&\displaystyle 3 y \left(- 10 x^{2} - 9 x y + 8 x - 2 y^{2} + 3 y - 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(1,0)\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x \left(10 x^{2} - 12 x + 3\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x \left(10 x^{2} - 12 x + 3\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(0,1)\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle y \left(10 y^{2} - 12 y + 3\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle y \left(10 y^{2} - 12 y + 3\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(0,1)\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 2 x^{2} - 9 x y + 3 x - 10 y^{2} + 8 y - 1\right)&\displaystyle y \left(18 x^{2} + 27 x y - 18 x + 10 y^{2} - 12 y + 3\right)\\\displaystyle y \left(18 x^{2} + 27 x y - 18 x + 10 y^{2} - 12 y + 3\right)&\displaystyle 9 y^{2} \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 2 x^{2} - 9 x y + 3 x - 10 y^{2} + 8 y - 1\right)&\displaystyle y \left(18 x^{2} + 27 x y - 18 x + 10 y^{2} - 12 y + 3\right)\\\displaystyle y \left(18 x^{2} + 27 x y - 18 x + 10 y^{2} - 12 y + 3\right)&\displaystyle 9 y^{2} \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,1)\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 3 x^{2} \left(- x - 2 y + 1\right)&\displaystyle 3 x y \left(3 x + 2 y - 2\right)\\\displaystyle 3 x y \left(3 x + 2 y - 2\right)&\displaystyle y^{2} \left(- 9 x - 2 y + 3\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 3 x^{2} \left(- x - 2 y + 1\right)&\displaystyle 3 x y \left(3 x + 2 y - 2\right)\\\displaystyle 3 x y \left(3 x + 2 y - 2\right)&\displaystyle y^{2} \left(- 9 x - 2 y + 3\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 3 x \left(- x^{2} + 3 x y + 2 x - 1\right)&\displaystyle 9 x y \left(x - y\right)\\\displaystyle 9 x y \left(x - y\right)&\displaystyle 3 y \left(- 3 x y + 4 x + y^{2} - 1\right)\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 3 x \left(- x^{2} + 3 x y + 2 x - 1\right)&\displaystyle 9 x y \left(x - y\right)\\\displaystyle 9 x y \left(x - y\right)&\displaystyle 3 y \left(- 3 x y + 4 x + y^{2} - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 3 x \left(5 x^{2} + 15 x y - 4 x - 1\right)&\displaystyle 9 x y \left(- 5 x - 5 y + 4\right)\\\displaystyle 9 x y \left(- 5 x - 5 y + 4\right)&\displaystyle 3 y \left(15 x y - 4 x + 5 y^{2} - 6 y + 1\right)\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 3 x \left(5 x^{2} + 15 x y - 4 x - 1\right)&\displaystyle 9 x y \left(- 5 x - 5 y + 4\right)\\\displaystyle 9 x y \left(- 5 x - 5 y + 4\right)&\displaystyle 3 y \left(15 x y - 4 x + 5 y^{2} - 6 y + 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 3 x \left(x^{2} - 3 x y + 4 y - 1\right)&\displaystyle 9 x y \left(- x + y\right)\\\displaystyle 9 x y \left(- x + y\right)&\displaystyle 3 y \left(3 x y - y^{2} + 2 y - 1\right)\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 3 x \left(x^{2} - 3 x y + 4 y - 1\right)&\displaystyle 9 x y \left(- x + y\right)\\\displaystyle 9 x y \left(- x + y\right)&\displaystyle 3 y \left(3 x y - y^{2} + 2 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 5 x^{2} - 15 x y + 6 x + 4 y - 1\right)&\displaystyle 9 x y \left(5 x + 5 y - 4\right)\\\displaystyle 9 x y \left(5 x + 5 y - 4\right)&\displaystyle 3 y \left(- 15 x y - 5 y^{2} + 4 y + 1\right)\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 5 x^{2} - 15 x y + 6 x + 4 y - 1\right)&\displaystyle 9 x y \left(5 x + 5 y - 4\right)\\\displaystyle 9 x y \left(5 x + 5 y - 4\right)&\displaystyle 3 y \left(- 15 x y - 5 y^{2} + 4 y + 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle - 12 x^{3} - 54 x^{2} y + 30 x^{2} + 24 x y - 18 x + 60 y^{3} - 96 y^{2} + 36 y&\displaystyle 18 x y \left(2 x + 3 y - 2\right)\\\displaystyle 18 x y \left(2 x + 3 y - 2\right)&\displaystyle 18 y^{2} \left(- 2 x - y + 1\right)\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle - 12 x^{3} - 54 x^{2} y + 30 x^{2} + 24 x y - 18 x + 60 y^{3} - 96 y^{2} + 36 y&\displaystyle 18 x y \left(2 x + 3 y - 2\right)\\\displaystyle 18 x y \left(2 x + 3 y - 2\right)&\displaystyle 18 y^{2} \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 6 x \left(5 x^{2} + 30 x y - 11 x + 30 y^{2} - 32 y + 6\right)&\displaystyle 6 y \left(- 15 x^{2} - 30 x y + 22 x - 10 y^{2} + 16 y - 6\right)\\\displaystyle 6 y \left(- 15 x^{2} - 30 x y + 22 x - 10 y^{2} + 16 y - 6\right)&\displaystyle 6 y \left(15 x y - 4 x + 10 y^{2} - 13 y + 3\right)\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 6 x \left(5 x^{2} + 30 x y - 11 x + 30 y^{2} - 32 y + 6\right)&\displaystyle 6 y \left(- 15 x^{2} - 30 x y + 22 x - 10 y^{2} + 16 y - 6\right)\\\displaystyle 6 y \left(- 15 x^{2} - 30 x y + 22 x - 10 y^{2} + 16 y - 6\right)&\displaystyle 6 y \left(15 x y - 4 x + 10 y^{2} - 13 y + 3\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle 12 x^{3} + 54 x^{2} y - 18 x^{2} - 24 x y + 6 x - 60 y^{3} + 84 y^{2} - 24 y&\displaystyle 18 x y \left(- 2 x - 3 y + 2\right)\\\displaystyle 18 x y \left(- 2 x - 3 y + 2\right)&\displaystyle 18 y^{2} \left(2 x + y - 1\right)\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle 12 x^{3} + 54 x^{2} y - 18 x^{2} - 24 x y + 6 x - 60 y^{3} + 84 y^{2} - 24 y&\displaystyle 18 x y \left(- 2 x - 3 y + 2\right)\\\displaystyle 18 x y \left(- 2 x - 3 y + 2\right)&\displaystyle 18 y^{2} \left(2 x + y - 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle 6 x \left(- 5 x^{2} - 30 x y + 9 x - 30 y^{2} + 28 y - 4\right)&\displaystyle 6 y \left(15 x^{2} + 30 x y - 18 x + 10 y^{2} - 14 y + 4\right)\\\displaystyle 6 y \left(15 x^{2} + 30 x y - 18 x + 10 y^{2} - 14 y + 4\right)&\displaystyle 6 y \left(- 15 x y - 10 y^{2} + 11 y - 1\right)\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle 6 x \left(- 5 x^{2} - 30 x y + 9 x - 30 y^{2} + 28 y - 4\right)&\displaystyle 6 y \left(15 x^{2} + 30 x y - 18 x + 10 y^{2} - 14 y + 4\right)\\\displaystyle 6 y \left(15 x^{2} + 30 x y - 18 x + 10 y^{2} - 14 y + 4\right)&\displaystyle 6 y \left(- 15 x y - 10 y^{2} + 11 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle 18 x^{2} \left(- x - 2 y + 1\right)&\displaystyle 18 x y \left(3 x + 2 y - 2\right)\\\displaystyle 18 x y \left(3 x + 2 y - 2\right)&\displaystyle 60 x^{3} - 96 x^{2} - 54 x y^{2} + 24 x y + 36 x - 12 y^{3} + 30 y^{2} - 18 y\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle 18 x^{2} \left(- x - 2 y + 1\right)&\displaystyle 18 x y \left(3 x + 2 y - 2\right)\\\displaystyle 18 x y \left(3 x + 2 y - 2\right)&\displaystyle 60 x^{3} - 96 x^{2} - 54 x y^{2} + 24 x y + 36 x - 12 y^{3} + 30 y^{2} - 18 y\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{cc}\displaystyle 6 x \left(- 10 x^{2} - 15 x y + 13 x + 4 y - 3\right)&\displaystyle 6 x \left(10 x^{2} + 30 x y - 16 x + 15 y^{2} - 22 y + 6\right)\\\displaystyle 6 x \left(10 x^{2} + 30 x y - 16 x + 15 y^{2} - 22 y + 6\right)&\displaystyle 6 y \left(- 30 x^{2} - 30 x y + 32 x - 5 y^{2} + 11 y - 6\right)\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{cc}\displaystyle 6 x \left(- 10 x^{2} - 15 x y + 13 x + 4 y - 3\right)&\displaystyle 6 x \left(10 x^{2} + 30 x y - 16 x + 15 y^{2} - 22 y + 6\right)\\\displaystyle 6 x \left(10 x^{2} + 30 x y - 16 x + 15 y^{2} - 22 y + 6\right)&\displaystyle 6 y \left(- 30 x^{2} - 30 x y + 32 x - 5 y^{2} + 11 y - 6\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{cc}\displaystyle 18 x^{2} \left(x + 2 y - 1\right)&\displaystyle 18 x y \left(- 3 x - 2 y + 2\right)\\\displaystyle 18 x y \left(- 3 x - 2 y + 2\right)&\displaystyle - 60 x^{3} + 84 x^{2} + 54 x y^{2} - 24 x y - 24 x + 12 y^{3} - 18 y^{2} + 6 y\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{cc}\displaystyle 18 x^{2} \left(x + 2 y - 1\right)&\displaystyle 18 x y \left(- 3 x - 2 y + 2\right)\\\displaystyle 18 x y \left(- 3 x - 2 y + 2\right)&\displaystyle - 60 x^{3} + 84 x^{2} + 54 x y^{2} - 24 x y - 24 x + 12 y^{3} - 18 y^{2} + 6 y\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{cc}\displaystyle 6 x \left(10 x^{2} + 15 x y - 11 x + 1\right)&\displaystyle 6 x \left(- 10 x^{2} - 30 x y + 14 x - 15 y^{2} + 18 y - 4\right)\\\displaystyle 6 x \left(- 10 x^{2} - 30 x y + 14 x - 15 y^{2} + 18 y - 4\right)&\displaystyle 6 y \left(30 x^{2} + 30 x y - 28 x + 5 y^{2} - 9 y + 4\right)\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{cc}\displaystyle 6 x \left(10 x^{2} + 15 x y - 11 x + 1\right)&\displaystyle 6 x \left(- 10 x^{2} - 30 x y + 14 x - 15 y^{2} + 18 y - 4\right)\\\displaystyle 6 x \left(- 10 x^{2} - 30 x y + 14 x - 15 y^{2} + 18 y - 4\right)&\displaystyle 6 y \left(30 x^{2} + 30 x y - 28 x + 5 y^{2} - 9 y + 4\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{21}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{cc}\displaystyle 24 x \left(- x - y + 1\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{cc}\displaystyle 24 x \left(- x - y + 1\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{22}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{cc}\displaystyle 24 x \left(- x - 2 y + 1\right)&\displaystyle 24 x y\\\displaystyle 24 x y&\displaystyle 24 y \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{cc}\displaystyle 24 x \left(- x - 2 y + 1\right)&\displaystyle 24 x y\\\displaystyle 24 x y&\displaystyle 24 y \left(- 2 x - y + 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{23}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right))v\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 24 y \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 24 y \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.