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Degree 3 reduced Hsieh–Clough–Tocher 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: \(1\), \(x\), \(y\), \(x^{2}\), \(x y\), \(y^{2}\), \(x^{3} - y^{3}\), \(\begin{cases} 4 x^{3} - 3 x y^{2} + 2 x y + 4 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + 12 x^{2} y - 7 x^{2} + 9 x y^{2} - 14 x y + 5 x + 4 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x^{3} + x^{2} - 2 y^{3} + 5 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\), \(\begin{cases} 25 x^{3} - 24 x y^{2} + 30 x y - 24 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\35 x^{3} + 33 x^{2} y - 21 x^{2} - 12 x y^{2} + 12 x - 28 y^{3} - 3 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x^{3} + 21 x^{2} y + 15 x^{2} - 23 y^{3} - 9 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} - 3 x y^{2} + 3 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + 21 x^{2} y - 15 x^{2} + 21 x y^{2} - 30 x y + 9 x + 7 y^{3} - 15 y^{2} + 9 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x^{3} - 3 x^{2} y - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} - 3 x y^{2} + 3 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + 21 x^{2} y - 15 x^{2} + 21 x y^{2} - 30 x y + 9 x + 7 y^{3} - 15 y^{2} + 9 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x^{3} - 3 x^{2} y - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)\)
\(\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} - \tfrac{3 x y^{2}}{2} + x + y^{3} - \tfrac{y^{2}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{5 x^{3}}{2} + 6 x^{2} y - \tfrac{11 x^{2}}{2} + \tfrac{9 x y^{2}}{2} - 8 x y + \tfrac{7 x}{2} + y^{3} - \tfrac{5 y^{2}}{2} + 2 y - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(x^{2} - 3 x - 2 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} - \tfrac{3 x y^{2}}{2} + x + y^{3} - \tfrac{y^{2}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{5 x^{3}}{2} + 6 x^{2} y - \tfrac{11 x^{2}}{2} + \tfrac{9 x y^{2}}{2} - 8 x y + \tfrac{7 x}{2} + y^{3} - \tfrac{5 y^{2}}{2} + 2 y - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(x^{2} - 3 x - 2 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)\)
\(\displaystyle \phi_{2} = \begin{cases} \tfrac{y \left(- 2 x + y^{2} - 3 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x^{3} + \tfrac{9 x^{2} y}{2} - \tfrac{5 x^{2}}{2} + 6 x y^{2} - 8 x y + 2 x + \tfrac{5 y^{3}}{2} - \tfrac{11 y^{2}}{2} + \tfrac{7 y}{2} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{3} - \tfrac{3 x^{2} y}{2} - \tfrac{x^{2}}{2} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{2} = \begin{cases} \tfrac{y \left(- 2 x + y^{2} - 3 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x^{3} + \tfrac{9 x^{2} y}{2} - \tfrac{5 x^{2}}{2} + 6 x y^{2} - 8 x y + 2 x + \tfrac{5 y^{3}}{2} - \tfrac{11 y^{2}}{2} + \tfrac{7 y}{2} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x^{3} - \tfrac{3 x^{2} y}{2} - \tfrac{x^{2}}{2} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} + \tfrac{3 x y^{2}}{2} - \tfrac{y^{3}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{9 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 9 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{5 y^{3}}{2} + 6 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(- 5 x + 3 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} + \tfrac{3 x y^{2}}{2} - \tfrac{y^{3}}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{9 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 9 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{5 y^{3}}{2} + 6 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(- 5 x + 3 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)\)
\(\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} - \tfrac{3 x y^{2}}{4} + \tfrac{y^{3}}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{9 x^{3}}{4} + \tfrac{21 x^{2} y}{4} - 4 x^{2} + \tfrac{21 x y^{2}}{4} - \tfrac{15 x y}{2} + \tfrac{9 x}{4} + \tfrac{5 y^{3}}{4} - 3 y^{2} + \tfrac{9 y}{4} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(5 x - 3 y - 4\right)}{4}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} - \tfrac{3 x y^{2}}{4} + \tfrac{y^{3}}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{9 x^{3}}{4} + \tfrac{21 x^{2} y}{4} - 4 x^{2} + \tfrac{21 x y^{2}}{4} - \tfrac{15 x y}{2} + \tfrac{9 x}{4} + \tfrac{5 y^{3}}{4} - 3 y^{2} + \tfrac{9 y}{4} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(5 x - 3 y - 4\right)}{4}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)\)
\(\displaystyle \phi_{5} = \begin{cases} \tfrac{y \left(- 3 x y + 4 x + 3 y^{2} - 2 y\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x^{3}}{4} + \tfrac{3 x^{2} y}{4} - \tfrac{x^{2}}{2} - \tfrac{3 x y^{2}}{4} + \tfrac{x y}{2} + \tfrac{x}{4} - \tfrac{y^{3}}{4} + \tfrac{y^{2}}{2} - \tfrac{y}{4}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(- 3 x + 3 y + 2\right)}{4}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{5} = \begin{cases} \tfrac{y \left(- 3 x y + 4 x + 3 y^{2} - 2 y\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x^{3}}{4} + \tfrac{3 x^{2} y}{4} - \tfrac{x^{2}}{2} - \tfrac{3 x y^{2}}{4} + \tfrac{x y}{2} + \tfrac{x}{4} - \tfrac{y^{3}}{4} + \tfrac{y^{2}}{2} - \tfrac{y}{4}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \left(- 3 x + 3 y + 2\right)}{4}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{6} = \begin{cases} \tfrac{y^{2} \left(3 x - 5 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{5 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 6 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{9 y^{3}}{2} + 9 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{x^{3}}{2} + \tfrac{3 x^{2} y}{2} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{6} = \begin{cases} \tfrac{y^{2} \left(3 x - 5 y + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{5 x^{3}}{2} - \tfrac{21 x^{2} y}{2} + 6 x^{2} - \tfrac{21 x y^{2}}{2} + 15 x y - \tfrac{9 x}{2} - \tfrac{9 y^{3}}{2} + 9 y^{2} - \tfrac{9 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{x^{3}}{2} + \tfrac{3 x^{2} y}{2} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)
\(\displaystyle \phi_{7} = \begin{cases} \tfrac{y^{2} \left(3 x - 3 y + 2\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{x^{3}}{4} - \tfrac{3 x^{2} y}{4} + \tfrac{x^{2}}{2} + \tfrac{3 x y^{2}}{4} + \tfrac{x y}{2} - \tfrac{x}{4} + \tfrac{y^{3}}{4} - \tfrac{y^{2}}{2} + \tfrac{y}{4}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(3 x^{2} - 3 x y - 2 x + 4 y\right)}{4}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{7} = \begin{cases} \tfrac{y^{2} \left(3 x - 3 y + 2\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{x^{3}}{4} - \tfrac{3 x^{2} y}{4} + \tfrac{x^{2}}{2} + \tfrac{3 x y^{2}}{4} + \tfrac{x y}{2} - \tfrac{x}{4} + \tfrac{y^{3}}{4} - \tfrac{y^{2}}{2} + \tfrac{y}{4}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(3 x^{2} - 3 x y - 2 x + 4 y\right)}{4}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)
\(\displaystyle \phi_{8} = \begin{cases} \tfrac{y^{2} \left(- 3 x + 5 y - 4\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{5 x^{3}}{4} + \tfrac{21 x^{2} y}{4} - 3 x^{2} + \tfrac{21 x y^{2}}{4} - \tfrac{15 x y}{2} + \tfrac{9 x}{4} + \tfrac{9 y^{3}}{4} - 4 y^{2} + \tfrac{9 y}{4} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{3}}{4} - \tfrac{3 x^{2} y}{4} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{8} = \begin{cases} \tfrac{y^{2} \left(- 3 x + 5 y - 4\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{5 x^{3}}{4} + \tfrac{21 x^{2} y}{4} - 3 x^{2} + \tfrac{21 x y^{2}}{4} - \tfrac{15 x y}{2} + \tfrac{9 x}{4} + \tfrac{9 y^{3}}{4} - 4 y^{2} + \tfrac{9 y}{4} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{3}}{4} - \tfrac{3 x^{2} y}{4} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
This DOF is associated with vertex 2 of the reference element.