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Reduced Hsieh–Clough–Tocher

 Abbreviated names rHCT Orders $$k=3$$ Reference elements triangle Number of DOFs triangle: $$9$$ Categories Scalar-valued elements

Implementations

 Symfem "rHCT"↓ Show Symfem examples ↓

Examples

triangle
order 3
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\begin{cases} 2 x^{3} - 3 x^{2} + 12 x y^{2} + 40 y^{3} - 24 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\40 x^{3} + 12 x^{2} y - 24 x^{2} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} x^{3} - 2 x^{2} + x + 8 y^{3} - 4 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(6 x^{2} + 3 x y - 5 x - y + 1\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y \left(3 x y - x + 6 y^{2} - 5 y + 1\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\8 x^{3} - 4 x^{2} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} - 2 x^{3} + 3 x^{2} - 12 x y^{2} + 14 y^{3} - 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 34 x^{3} - 87 x^{2} y + 81 x^{2} - 75 x y^{2} + 135 x y - 60 x - 20 y^{3} + 54 y^{2} - 48 y + 14&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} x^{3} - x^{2} + 3 x y^{2} - 4 y^{3} + y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\11 x^{3} + \tfrac{57 x^{2} y}{2} - \tfrac{51 x^{2}}{2} + \tfrac{51 x y^{2}}{2} - \tfrac{89 x y}{2} + 19 x + 7 y^{3} - \tfrac{37 y^{2}}{2} + 16 y - \tfrac{9}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y \left(- 3 x y + x + 3 y^{2} - y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 5 x^{3} - \tfrac{27 x^{2} y}{2} + \tfrac{25 x^{2}}{2} - \tfrac{27 x y^{2}}{2} + \tfrac{47 x y}{2} - 10 x - 4 y^{3} + \tfrac{21 y^{2}}{2} - 9 y + \tfrac{5}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 20 x^{3} - 75 x^{2} y + 54 x^{2} - 87 x y^{2} + 135 x y - 48 x - 34 y^{3} + 81 y^{2} - 60 y + 14&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\14 x^{3} - 12 x^{2} y - 3 x^{2} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 4 x^{3} - \tfrac{27 x^{2} y}{2} + \tfrac{21 x^{2}}{2} - \tfrac{27 x y^{2}}{2} + \tfrac{47 x y}{2} - 9 x - 5 y^{3} + \tfrac{25 y^{2}}{2} - 10 y + \tfrac{5}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(3 x^{2} - 3 x y - x + y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + \tfrac{51 x^{2} y}{2} - \tfrac{37 x^{2}}{2} + \tfrac{57 x y^{2}}{2} - \tfrac{89 x y}{2} + 16 x + 11 y^{3} - \tfrac{51 y^{2}}{2} + 19 y - \tfrac{9}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 4 x^{3} + 3 x^{2} y + x^{2} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} + 12 x y^{2} + 40 y^{3} - 24 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\40 x^{3} + 12 x^{2} y - 24 x^{2} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)$$

$$\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} + x + 8 y^{3} - 4 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(6 x^{2} + 3 x y - 5 x - y + 1\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)$$

$$\displaystyle \phi_{2} = \begin{cases} y \left(3 x y - x + 6 y^{2} - 5 y + 1\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\8 x^{3} - 4 x^{2} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} - 12 x y^{2} + 14 y^{3} - 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 34 x^{3} - 87 x^{2} y + 81 x^{2} - 75 x y^{2} + 135 x y - 60 x - 20 y^{3} + 54 y^{2} - 48 y + 14&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)$$

$$\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} + 3 x y^{2} - 4 y^{3} + y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\11 x^{3} + \tfrac{57 x^{2} y}{2} - \tfrac{51 x^{2}}{2} + \tfrac{51 x y^{2}}{2} - \tfrac{89 x y}{2} + 19 x + 7 y^{3} - \tfrac{37 y^{2}}{2} + 16 y - \tfrac{9}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)$$

$$\displaystyle \phi_{5} = \begin{cases} y \left(- 3 x y + x + 3 y^{2} - y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 5 x^{3} - \tfrac{27 x^{2} y}{2} + \tfrac{25 x^{2}}{2} - \tfrac{27 x y^{2}}{2} + \tfrac{47 x y}{2} - 10 x - 4 y^{3} + \tfrac{21 y^{2}}{2} - 9 y + \tfrac{5}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{6} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 20 x^{3} - 75 x^{2} y + 54 x^{2} - 87 x y^{2} + 135 x y - 48 x - 34 y^{3} + 81 y^{2} - 60 y + 14&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\14 x^{3} - 12 x^{2} y - 3 x^{2} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)$$

$$\displaystyle \phi_{7} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 4 x^{3} - \tfrac{27 x^{2} y}{2} + \tfrac{21 x^{2}}{2} - \tfrac{27 x y^{2}}{2} + \tfrac{47 x y}{2} - 9 x - 5 y^{3} + \tfrac{25 y^{2}}{2} - 10 y + \tfrac{5}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(3 x^{2} - 3 x y - x + y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)$$

$$\displaystyle \phi_{8} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\7 x^{3} + \tfrac{51 x^{2} y}{2} - \tfrac{37 x^{2}}{2} + \tfrac{57 x y^{2}}{2} - \tfrac{89 x y}{2} + 16 x + 11 y^{3} - \tfrac{51 y^{2}}{2} + 19 y - \tfrac{9}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 4 x^{3} + 3 x^{2} y + x^{2} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 2 of the reference element.

References

• Ciarlet, Philippe G. Interpolation error estimates for the reduced Hsieh-Clough-Tocher triangle, Mathematics of Computation 32, 335–344, 1978. [DOI: 10.1090/S0025-5718-1978-0482249-1] [BibTeX]

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 Element added 08 March 2021 Element last updated 10 February 2022