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Degree 2 Hellan–Herrmann–Johnson 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: \(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{2}\\\displaystyle x^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x y\\\displaystyle x y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y^{2}\\\displaystyle y^{2}&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{17}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 15 x^{2} - 12 x + \frac{3}{2}\\\displaystyle 15 x^{2} - 12 x + \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 15 x^{2} - 12 x + \frac{3}{2}\\\displaystyle 15 x^{2} - 12 x + \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(s_{0} \left(2 s_{0} - 1\right))|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 15 y^{2} - 12 y + \frac{3}{2}\\\displaystyle 15 y^{2} - 12 y + \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 15 y^{2} - 12 y + \frac{3}{2}\\\displaystyle 15 y^{2} - 12 y + \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(4 s_{0} \left(1 - s_{0}\right))|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}\\\displaystyle \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}\\\displaystyle \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9&\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9&\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(s_{0} \left(2 s_{0} - 1\right))|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 30 y^{2} - 24 y + 3&\displaystyle - 15 y^{2} + 12 y - \frac{3}{2}\\\displaystyle - 15 y^{2} + 12 y - \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 30 y^{2} - 24 y + 3&\displaystyle - 15 y^{2} + 12 y - \frac{3}{2}\\\displaystyle - 15 y^{2} + 12 y - \frac{3}{2}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(4 s_{0} \left(1 - s_{0}\right))|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle \frac{15 x^{2}}{2} - 15 x y - 3 x - 15 y^{2} + 15 y - \frac{3}{2}&\displaystyle - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}\\\displaystyle - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle \frac{15 x^{2}}{2} - 15 x y - 3 x - 15 y^{2} + 15 y - \frac{3}{2}&\displaystyle - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}\\\displaystyle - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}&\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}&\displaystyle 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\\displaystyle - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}&\displaystyle 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(s_{0} \left(2 s_{0} - 1\right))|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 15 x^{2} + 12 x - \frac{3}{2}\\\displaystyle - 15 x^{2} + 12 x - \frac{3}{2}&\displaystyle 30 x^{2} - 24 x + 3\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 15 x^{2} + 12 x - \frac{3}{2}\\\displaystyle - 15 x^{2} + 12 x - \frac{3}{2}&\displaystyle 30 x^{2} - 24 x + 3\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(4 s_{0} \left(1 - s_{0}\right))|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}\\\displaystyle \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}&\displaystyle - 15 x^{2} - 15 x y + 15 x + \frac{15 y^{2}}{2} - 3 y - \frac{3}{2}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}\\\displaystyle \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}&\displaystyle - 15 x^{2} - 15 x y + 15 x + \frac{15 y^{2}}{2} - 3 y - \frac{3}{2}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle - s_{0} - s_{1} + 1\\\displaystyle - s_{0} - s_{1} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 12 x \left(- 5 x - 5 y + 4\right)&\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36\\\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36&\displaystyle 12 y \left(- 5 x - 5 y + 4\right)\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 12 x \left(- 5 x - 5 y + 4\right)&\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36\\\displaystyle 90 x^{2} + 180 x y - 120 x + 90 y^{2} - 120 y + 36&\displaystyle 12 y \left(- 5 x - 5 y + 4\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 2 s_{0} + 2 s_{1} - 2&\displaystyle - s_{0} - s_{1} + 1\\\displaystyle - s_{0} - s_{1} + 1&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 12 x \left(5 x + 5 y - 4\right)&\displaystyle - 45 x^{2} - 60 x y + 48 x - 15 y^{2} + 24 y - 9\\\displaystyle - 45 x^{2} - 60 x y + 48 x - 15 y^{2} + 24 y - 9&\displaystyle 0\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 12 x \left(5 x + 5 y - 4\right)&\displaystyle - 45 x^{2} - 60 x y + 48 x - 15 y^{2} + 24 y - 9\\\displaystyle - 45 x^{2} - 60 x y + 48 x - 15 y^{2} + 24 y - 9&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle s_{0} + s_{1} - 1\\\displaystyle s_{0} + s_{1} - 1&\displaystyle - 2 s_{0} - 2 s_{1} + 2\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 15 x^{2} + 60 x y - 24 x + 45 y^{2} - 48 y + 9\\\displaystyle 15 x^{2} + 60 x y - 24 x + 45 y^{2} - 48 y + 9&\displaystyle 12 y \left(- 5 x - 5 y + 4\right)\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 15 x^{2} + 60 x y - 24 x + 45 y^{2} - 48 y + 9\\\displaystyle 15 x^{2} + 60 x y - 24 x + 45 y^{2} - 48 y + 9&\displaystyle 12 y \left(- 5 x - 5 y + 4\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{12}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 6 x \left(5 x - 2\right)&\displaystyle - 135 x^{2} - 150 x y + 150 x + 30 y - 24\\\displaystyle - 135 x^{2} - 150 x y + 150 x + 30 y - 24&\displaystyle 12 y \left(5 x - 1\right)\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 6 x \left(5 x - 2\right)&\displaystyle - 135 x^{2} - 150 x y + 150 x + 30 y - 24\\\displaystyle - 135 x^{2} - 150 x y + 150 x + 30 y - 24&\displaystyle 12 y \left(5 x - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle - 2 s_{0}&\displaystyle s_{0}\\\displaystyle s_{0}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle 6 x \left(2 - 5 x\right)&\displaystyle 45 x^{2} + 30 x y - 42 x - 6 y + 6\\\displaystyle 45 x^{2} + 30 x y - 42 x - 6 y + 6&\displaystyle 0\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle 6 x \left(2 - 5 x\right)&\displaystyle 45 x^{2} + 30 x y - 42 x - 6 y + 6\\\displaystyle 45 x^{2} + 30 x y - 42 x - 6 y + 6&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle - s_{0}\\\displaystyle - s_{0}&\displaystyle 2 s_{0}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y + 36 x + 12 y - 6\\\displaystyle - 30 x^{2} - 60 x y + 36 x + 12 y - 6&\displaystyle 12 y \left(5 x - 1\right)\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 30 x^{2} - 60 x y + 36 x + 12 y - 6\\\displaystyle - 30 x^{2} - 60 x y + 36 x + 12 y - 6&\displaystyle 12 y \left(5 x - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle 12 x \left(5 y - 1\right)&\displaystyle - 150 x y + 30 x - 135 y^{2} + 150 y - 24\\\displaystyle - 150 x y + 30 x - 135 y^{2} + 150 y - 24&\displaystyle 6 y \left(5 y - 2\right)\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle 12 x \left(5 y - 1\right)&\displaystyle - 150 x y + 30 x - 135 y^{2} + 150 y - 24\\\displaystyle - 150 x y + 30 x - 135 y^{2} + 150 y - 24&\displaystyle 6 y \left(5 y - 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{16}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle - 2 s_{1}&\displaystyle s_{1}\\\displaystyle s_{1}&\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle 12 x \left(1 - 5 y\right)&\displaystyle 60 x y - 12 x + 30 y^{2} - 36 y + 6\\\displaystyle 60 x y - 12 x + 30 y^{2} - 36 y + 6&\displaystyle 0\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle 12 x \left(1 - 5 y\right)&\displaystyle 60 x y - 12 x + 30 y^{2} - 36 y + 6\\\displaystyle 60 x y - 12 x + 30 y^{2} - 36 y + 6&\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle - s_{1}\\\displaystyle - s_{1}&\displaystyle 2 s_{1}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 30 x y + 6 x - 45 y^{2} + 42 y - 6\\\displaystyle - 30 x y + 6 x - 45 y^{2} + 42 y - 6&\displaystyle 6 y \left(5 y - 2\right)\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 30 x y + 6 x - 45 y^{2} + 42 y - 6\\\displaystyle - 30 x y + 6 x - 45 y^{2} + 42 y - 6&\displaystyle 6 y \left(5 y - 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.