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Degree 1 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: \(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle x&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle y&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{11}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{28 x}{15} - \frac{8 y}{15} - \frac{1}{3}&\displaystyle - \frac{4 y}{15}\\\displaystyle - \frac{14 x}{15}&\displaystyle - \frac{28 x}{15} + \frac{8 y}{15} + \frac{1}{3}\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{28 x}{15} - \frac{8 y}{15} - \frac{1}{3}&\displaystyle - \frac{4 y}{15}\\\displaystyle - \frac{14 x}{15}&\displaystyle - \frac{28 x}{15} + \frac{8 y}{15} + \frac{1}{3}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle - \frac{8 x}{15} + \frac{28 y}{15} - \frac{1}{3}&\displaystyle \frac{14 y}{15}\\\displaystyle \frac{4 x}{15}&\displaystyle \frac{8 x}{15} - \frac{28 y}{15} + \frac{1}{3}\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle - \frac{8 x}{15} + \frac{28 y}{15} - \frac{1}{3}&\displaystyle \frac{14 y}{15}\\\displaystyle \frac{4 x}{15}&\displaystyle \frac{8 x}{15} - \frac{28 y}{15} + \frac{1}{3}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle \frac{4 x}{5} + \frac{6 y}{5} - \frac{1}{2}&\displaystyle \frac{3 y}{5}\\\displaystyle \frac{23 x}{5} + 6 y - 4&\displaystyle - \frac{4 x}{5} - \frac{6 y}{5} + \frac{1}{2}\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle \frac{4 x}{5} + \frac{6 y}{5} - \frac{1}{2}&\displaystyle \frac{3 y}{5}\\\displaystyle \frac{23 x}{5} + 6 y - 4&\displaystyle - \frac{4 x}{5} - \frac{6 y}{5} + \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle \frac{1}{2} - 2 y&\displaystyle - y\\\displaystyle - x - 6 y + 2&\displaystyle 2 y - \frac{1}{2}\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle \frac{1}{2} - 2 y&\displaystyle - y\\\displaystyle - x - 6 y + 2&\displaystyle 2 y - \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle \frac{6 x}{5} + \frac{4 y}{5} - \frac{1}{2}&\displaystyle - 6 x - \frac{23 y}{5} + 4\\\displaystyle - \frac{3 x}{5}&\displaystyle - \frac{6 x}{5} - \frac{4 y}{5} + \frac{1}{2}\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle \frac{6 x}{5} + \frac{4 y}{5} - \frac{1}{2}&\displaystyle - 6 x - \frac{23 y}{5} + 4\\\displaystyle - \frac{3 x}{5}&\displaystyle - \frac{6 x}{5} - \frac{4 y}{5} + \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle \frac{1}{2} - 2 x&\displaystyle 6 x + y - 2\\\displaystyle x&\displaystyle 2 x - \frac{1}{2}\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle \frac{1}{2} - 2 x&\displaystyle 6 x + y - 2\\\displaystyle x&\displaystyle 2 x - \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(- s_{0} - s_{1} + 1)\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle - 12 x - 12 y + 9&\displaystyle 0\\\displaystyle 0&\displaystyle - 12 x - 12 y + 9\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle - 12 x - 12 y + 9&\displaystyle 0\\\displaystyle 0&\displaystyle - 12 x - 12 y + 9\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(s_{0})\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 12 x - 3&\displaystyle 0\\\displaystyle 0&\displaystyle 12 x - 3\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 12 x - 3&\displaystyle 0\\\displaystyle 0&\displaystyle 12 x - 3\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})(s_{1})\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 12 y - 3&\displaystyle 0\\\displaystyle 0&\displaystyle 12 y - 3\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 12 y - 3&\displaystyle 0\\\displaystyle 0&\displaystyle 12 y - 3\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{9}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0}}{2} + \frac{s_{1}}{2} - \frac{1}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{s_{0}}{2} - \frac{s_{1}}{2} + \frac{1}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 16 x + 16 y - 14&\displaystyle - 4 y\\\displaystyle 4 x&\displaystyle - 16 x - 16 y + 14\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 16 x + 16 y - 14&\displaystyle - 4 y\\\displaystyle 4 x&\displaystyle - 16 x - 16 y + 14\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{0}}{2}&\displaystyle 0\\\displaystyle s_{0}&\displaystyle - \frac{s_{0}}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle \frac{32 x}{5} + \frac{8 y}{5} - 2&\displaystyle \frac{4 y}{5}\\\displaystyle \frac{44 x}{5}&\displaystyle - \frac{32 x}{5} - \frac{8 y}{5} + 2\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle \frac{32 x}{5} + \frac{8 y}{5} - 2&\displaystyle \frac{4 y}{5}\\\displaystyle \frac{44 x}{5}&\displaystyle - \frac{32 x}{5} - \frac{8 y}{5} + 2\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle \frac{s_{1}}{2}&\displaystyle - s_{1}\\\displaystyle 0&\displaystyle - \frac{s_{1}}{2}\end{array}\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle \frac{8 x}{5} + \frac{32 y}{5} - 2&\displaystyle - \frac{44 y}{5}\\\displaystyle - \frac{4 x}{5}&\displaystyle - \frac{8 x}{5} - \frac{32 y}{5} + 2\end{array}\right)\)
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 \(R\).
\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle \frac{8 x}{5} + \frac{32 y}{5} - 2&\displaystyle - \frac{44 y}{5}\\\displaystyle - \frac{4 x}{5}&\displaystyle - \frac{8 x}{5} - \frac{32 y}{5} + 2\end{array}\right)\)
This DOF is associated with face 0 of the reference element.