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Degree 1 vector bubble enriched Lagrange 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y \left(- x - y + 1\right)\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{7}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x \left(9 x y + 9 y^{2} - 9 y + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x \left(9 x y + 9 y^{2} - 9 y + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(9 x y + 9 y^{2} - 9 y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(9 x y + 9 y^{2} - 9 y + 1\right)\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(9 x^{2} + 9 x y - 9 x + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(9 x^{2} + 9 x y - 9 x + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(9 x^{2} + 9 x y - 9 x + 1\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(9 x^{2} + 9 x y - 9 x + 1\right)\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 27 x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 27 x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 27 x y \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 27 x y \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.