an encyclopedia of finite element definitions

# Hsieh–Clough–Tocher

 Alternative names Clough–Tocher Abbreviated names HCT, CT Orders $$k=3$$ Reference elements triangle Number of DOFs triangle: $$12$$ Categories Scalar-valued elements

## Implementations

 Symfem "HCT"↓ 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: $$\begin{cases} 2 x^{3} + 33 x^{2} y - 3 x^{2} + 45 x y^{2} - 33 x y - 26 y^{3} + 9 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 26 x^{3} + 45 x^{2} y + 9 x^{2} + 33 x y^{2} - 33 x y + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} x^{3} + 8 x^{2} y - 2 x^{2} + 8 x y^{2} - 8 x y + x - 8 y^{3} + 4 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(2 x^{2} + 5 x y - 3 x + 2 y^{2} - 3 y + 1\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y \left(2 x^{2} + 5 x y - 3 x + 2 y^{2} - 3 y + 1\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 8 x^{3} + 8 x^{2} y + 4 x^{2} + 8 x y^{2} - 8 x y + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} - 2 x^{3} + 27 x^{2} y + 3 x^{2} + 15 x y^{2} - 27 x y - 40 y^{3} + 24 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\26 x^{3} + 123 x^{2} y - 69 x^{2} + 135 x y^{2} - 195 x y + 60 x + 40 y^{3} - 96 y^{2} + 72 y - 16&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} x^{3} - 7 x^{2} y - x^{2} - 4 x y^{2} + 7 x y + 10 y^{3} - 6 y^{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 6 x^{3} - 31 x^{2} y + 17 x^{2} - 34 x y^{2} + 49 x y - 15 x - 10 y^{3} + 24 y^{2} - 18 y + 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y \left(2 x^{2} - x y - x - y^{2} + y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\8 x^{3} + 32 x^{2} y - 20 x^{2} + 32 x y^{2} - 48 x y + 16 x + 9 y^{3} - 22 y^{2} + 17 y - 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 3 y \left(7 x^{2} + 7 x y - 7 x - 32 y^{2} + 16 y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\96 x^{3} + 309 x^{2} y - 240 x^{2} + 309 x y^{2} - 501 x y + 192 x + 96 y^{3} - 240 y^{2} + 192 y - 48&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x \left(- 32 x^{2} + 7 x y + 16 x + 7 y^{2} - 7 y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y \left(x^{2} + 10 x y - x - 2 y^{2} - 2 y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\11 x^{3} + 25 x^{2} y - 26 x^{2} + 16 x y^{2} - 35 x y + 19 x + 2 y^{3} - 8 y^{2} + 10 y - 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(13 x^{2} - 2 x y - 5 x - 2 y^{2} + 2 y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y \left(- 2 x^{2} - 2 x y + 2 x + 13 y^{2} - 5 y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\2 x^{3} + 16 x^{2} y - 8 x^{2} + 25 x y^{2} - 35 x y + 10 x + 11 y^{3} - 26 y^{2} + 19 y - 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(- 2 x^{2} + 10 x y - 2 x + y^{2} - y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\40 x^{3} + 135 x^{2} y - 96 x^{2} + 123 x y^{2} - 195 x y + 72 x + 26 y^{3} - 69 y^{2} + 60 y - 16&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 40 x^{3} + 15 x^{2} y + 24 x^{2} + 27 x y^{2} - 27 x y - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\9 x^{3} + 32 x^{2} y - 22 x^{2} + 32 x y^{2} - 48 x y + 17 x + 8 y^{3} - 20 y^{2} + 16 y - 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x \left(- x^{2} - x y + x + 2 y^{2} - y\right)&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 10 x^{3} - 34 x^{2} y + 24 x^{2} - 31 x y^{2} + 49 x y - 18 x - 6 y^{3} + 17 y^{2} - 15 y + 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\10 x^{3} - 4 x^{2} y - 6 x^{2} - 7 x y^{2} + 7 x y + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = \begin{cases} 2 x^{3} - 3 x^{2} + 111 x y^{2} + \tfrac{1369 y^{3}}{7} - \tfrac{828 y^{2}}{7} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{297 x^{3}}{7} + \tfrac{891 x^{2} y}{7} - \tfrac{858 x^{2}}{7} + \tfrac{891 x y^{2}}{7} - \tfrac{1716 x y}{7} + \tfrac{825 x}{7} + \tfrac{297 y^{3}}{7} - \tfrac{858 y^{2}}{7} + \tfrac{825 y}{7} - \tfrac{264}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{1369 x^{3}}{7} + 111 x^{2} y - \tfrac{828 x^{2}}{7} + 2 y^{3} - 3 y^{2} + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{1} = \begin{cases} x^{3} - 2 x^{2} + x + \tfrac{272 y^{3}}{7} - \tfrac{108 y^{2}}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{48 x^{3}}{7} + \tfrac{24 x^{2} y}{7} + \tfrac{64 x^{2}}{7} + \tfrac{192 x y^{2}}{7} - \tfrac{208 x y}{7} + \tfrac{16 x}{7} + \tfrac{120 y^{3}}{7} - \tfrac{272 y^{2}}{7} + \tfrac{184 y}{7} - \tfrac{32}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(62 x^{2} + 203 x y - 101 x + 14 y^{2} - 21 y + 7\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{2} = \begin{cases} \tfrac{y \left(14 x^{2} + 203 x y - 21 x + 62 y^{2} - 101 y + 7\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{120 x^{3}}{7} + \tfrac{192 x^{2} y}{7} - \tfrac{272 x^{2}}{7} + \tfrac{24 x y^{2}}{7} - \tfrac{208 x y}{7} + \tfrac{184 x}{7} - \tfrac{48 y^{3}}{7} + \tfrac{64 y^{2}}{7} + \tfrac{16 y}{7} - \tfrac{32}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{272 x^{3}}{7} - \tfrac{108 x^{2}}{7} + y^{3} - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{3} = \begin{cases} - 2 x^{3} + 3 x^{2} - 102 x y^{2} + \tfrac{377 y^{3}}{7} + \tfrac{39 y^{2}}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{1210 x^{3}}{7} - \tfrac{2958 x^{2} y}{7} + \tfrac{2913 x^{2}}{7} - \tfrac{2307 x y^{2}}{7} + \tfrac{4503 x y}{7} - \tfrac{2196 x}{7} - \tfrac{545 y^{3}}{7} + \tfrac{1590 y^{2}}{7} - \tfrac{1545 y}{7} + \tfrac{500}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{3 x^{2} \left(- 96 x - 21 y + 20\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{4} = \begin{cases} x^{3} - x^{2} + \tfrac{51 x y^{2}}{2} - \tfrac{209 y^{3}}{14} - \tfrac{11 y^{2}}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{615 x^{3}}{14} + \tfrac{737 x^{2} y}{7} - \tfrac{732 x^{2}}{7} + \tfrac{569 x y^{2}}{7} - \tfrac{2235 x y}{14} + \tfrac{1097 x}{14} + \tfrac{265 y^{3}}{14} - \tfrac{389 y^{2}}{7} + \tfrac{761 y}{14} - \tfrac{124}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x^{2} \cdot \left(141 x + 21 y - 25\right)}{14}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{5} = \begin{cases} \tfrac{y \left(28 x^{2} - 329 x y - 14 x - 239 y^{2} + 179 y\right)}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{503 x^{3}}{14} - \tfrac{751 x^{2} y}{7} + \tfrac{655 x^{2}}{7} - \tfrac{751 x y^{2}}{7} + \tfrac{2613 x y}{14} - \tfrac{1111 x}{14} - \tfrac{489 y^{3}}{14} + \tfrac{641 y^{2}}{7} - \tfrac{1097 y}{14} + \tfrac{152}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{15 x^{2} \left(- 15 x - 21 y + 11\right)}{14}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{6} = \begin{cases} \tfrac{3 y^{2} \left(- 21 x - 96 y + 20\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{545 x^{3}}{7} - \tfrac{2307 x^{2} y}{7} + \tfrac{1590 x^{2}}{7} - \tfrac{2958 x y^{2}}{7} + \tfrac{4503 x y}{7} - \tfrac{1545 x}{7} - \tfrac{1210 y^{3}}{7} + \tfrac{2913 y^{2}}{7} - \tfrac{2196 y}{7} + \tfrac{500}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{377 x^{3}}{7} - 102 x^{2} y + \tfrac{39 x^{2}}{7} - 2 y^{3} + 3 y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{7} = \begin{cases} \tfrac{15 y^{2} \left(- 21 x - 15 y + 11\right)}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{489 x^{3}}{14} - \tfrac{751 x^{2} y}{7} + \tfrac{641 x^{2}}{7} - \tfrac{751 x y^{2}}{7} + \tfrac{2613 x y}{14} - \tfrac{1097 x}{14} - \tfrac{503 y^{3}}{14} + \tfrac{655 y^{2}}{7} - \tfrac{1111 y}{14} + \tfrac{152}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{x \left(- 239 x^{2} - 329 x y + 179 x + 28 y^{2} - 14 y\right)}{14}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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

$$\displaystyle \phi_{8} = \begin{cases} \tfrac{y^{2} \cdot \left(21 x + 141 y - 25\right)}{14}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{265 x^{3}}{14} + \tfrac{569 x^{2} y}{7} - \tfrac{389 x^{2}}{7} + \tfrac{737 x y^{2}}{7} - \tfrac{2235 x y}{14} + \tfrac{761 x}{14} + \tfrac{615 y^{3}}{14} - \tfrac{732 y^{2}}{7} + \tfrac{1097 y}{14} - \tfrac{124}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{209 x^{3}}{14} + \tfrac{51 x^{2} y}{2} - \tfrac{11 x^{2}}{14} + y^{3} - y^{2}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}$$
where $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \phi_{9} = \begin{cases} \tfrac{2 \sqrt{2} y^{2} \left(- 21 x - 15 y + 11\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{2 \sqrt{2} \left(- 41 x^{3} - 130 x^{2} y + 106 x^{2} - 130 x y^{2} + 219 x y - 89 x - 41 y^{3} + 106 y^{2} - 89 y + 24\right)}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{2 \sqrt{2} x^{2} \left(- 15 x - 21 y + 11\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{10}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}$$
where $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

$$\displaystyle \phi_{10} = \begin{cases} \tfrac{4 y^{2} \left(- 21 x - 6 y + 10\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{60 x^{3}}{7} - \tfrac{96 x^{2} y}{7} + \tfrac{136 x^{2}}{7} - \tfrac{12 x y^{2}}{7} + \tfrac{104 x y}{7} - \tfrac{92 x}{7} + \tfrac{24 y^{3}}{7} - \tfrac{32 y^{2}}{7} - \tfrac{8 y}{7} + \tfrac{16}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{4 x \left(- 41 x^{2} + 7 x y + 17 x + 7 y^{2} - 7 y\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{11}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}$$
where $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \phi_{11} = \begin{cases} \tfrac{4 y \left(- 7 x^{2} - 7 x y + 7 x + 41 y^{2} - 17 y\right)}{7}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{24 x^{3}}{7} + \tfrac{12 x^{2} y}{7} + \tfrac{32 x^{2}}{7} + \tfrac{96 x y^{2}}{7} - \tfrac{104 x y}{7} + \tfrac{8 x}{7} + \tfrac{60 y^{3}}{7} - \tfrac{136 y^{2}}{7} + \tfrac{92 y}{7} - \tfrac{16}{7}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\\tfrac{4 x^{2} \cdot \left(6 x + 21 y - 10\right)}{7}&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

This DOF is associated with edge 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]

## DefElement stats

 Element added 08 March 2021 Element last updated 10 February 2022