an encyclopedia of finite element definitions

# Nonconforming Arnold–Winther

 Orders $$k=2$$ Reference elements triangle Polynomial set $$\mathcal{Z}^{(12)}_{k-1}$$↓ Show polynomial set definitions ↓ DOFs On each edge: integral moments of normal-normal and normal-tangent inner products with an order $$1$$ Lagrange space On each face: , and integral moments of three components with an order $$0$$ Lagrange space Number of DOFs triangle: $$15$$ Categories Matrix-valued elements

## Implementations

 Symfem "nonconforming AW"↓ Show Symfem examples ↓ UFL "AWnc"↓ Show UFL examples ↓

## Examples

triangle
order 2
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)$$, $$\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)$$, $$\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)$$, $$\left(\begin{matrix}0 & y^{2}\\y^{2} & - 2 y^{2}\end{matrix}\right)$$, $$\left(\begin{matrix}- 2 x^{2} & x^{2}\\x^{2} & 0\end{matrix}\right)$$, $$\left(\begin{matrix}- 2 x y & x y\\x y & 0\end{matrix}\right)$$, $$\left(\begin{matrix}x \left(x - y\right) & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}x^{2} & 0\\0 & x y\end{matrix}\right)$$, $$\left(\begin{matrix}x^{2} & 0\\0 & y^{2}\end{matrix}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{14}\}$$
• 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 $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}\frac{3 x \left(3 x - 2 y - 1\right)}{2} & \frac{3 x^{2}}{2} + 3 x y - \frac{3 x}{2} + \frac{3 y^{2}}{2} - \frac{3 y}{2} + \frac{1}{4}\\\frac{3 x^{2}}{2} + 3 x y - \frac{3 x}{2} + \frac{3 y^{2}}{2} - \frac{3 y}{2} + \frac{1}{4} & \frac{3 y \left(6 x - y - 1\right)}{2}\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}\frac{3 x \left(7 x + 6 y - 5\right)}{2} & - \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4}\\- \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4} & \frac{3 y \left(- 2 x + 3 y - 1\right)}{2}\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}\frac{3 x \left(x + 10 y - 3\right)}{2} & - \frac{3 x^{2}}{2} - 3 x y + \frac{3 x}{2} - \frac{3 y^{2}}{2} + \frac{3 y}{2} - \frac{1}{4}\\- \frac{3 x^{2}}{2} - 3 x y + \frac{3 x}{2} - \frac{3 y^{2}}{2} + \frac{3 y}{2} - \frac{1}{4} & \frac{3 y \left(2 x + 5 y - 3\right)}{2}\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}\frac{3 x \left(3 x + 14 y - 5\right)}{2} & - \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4}\\- \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4} & \frac{3 y \left(6 x - y - 1\right)}{2}\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}21 x^{2} + 42 x y - 27 x - 6 y + 4 & - 9 x^{2} - 18 x y + 9 x - 9 y^{2} + 9 y - \frac{3}{2}\\- 9 x^{2} - 18 x y + 9 x - 9 y^{2} + 9 y - \frac{3}{2} & 9 y \left(2 x + y - 1\right)\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}3 x \left(- 5 x - 18 y + 7\right) & 3 x^{2} + 30 x y - 3 x + 27 y^{2} - 21 y + \frac{1}{2}\\3 x^{2} + 30 x y - 3 x + 27 y^{2} - 21 y + \frac{1}{2} & 3 y \left(- 10 x - 9 y + 7\right)\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}- 6 x^{2} - 36 x y + 12 x + 6 y - 2 & 6 x^{2} + 12 x y - 6 x + 6 y^{2} - 6 y + 1\\6 x^{2} + 12 x y - 6 x + 6 y^{2} - 6 y + 1 & 6 y \left(- 2 x - y + 1\right)\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}3 x \left(3 x + 14 y - 5\right) & - 9 x^{2} - 18 x y + 9 x - 21 y^{2} + 15 y - \frac{3}{2}\\- 9 x^{2} - 18 x y + 9 x - 21 y^{2} + 15 y - \frac{3}{2} & 3 y \left(6 x + 7 y - 5\right)\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}9 x \left(- x - 2 y + 1\right) & 9 x^{2} + 18 x y - 9 x + 9 y^{2} - 9 y + \frac{3}{2}\\9 x^{2} + 18 x y - 9 x + 9 y^{2} - 9 y + \frac{3}{2} & 6 x y - 6 x + 3 y^{2} - 9 y + 4\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{9} = \left(\begin{matrix}3 x \left(- x - 10 y + 3\right) & 3 x^{2} + 30 x y - 9 x + 27 y^{2} - 27 y + \frac{9}{2}\\3 x^{2} + 30 x y - 9 x + 27 y^{2} - 27 y + \frac{9}{2} & 3 y \left(- 2 x - 5 y + 3\right)\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{10} = \left(\begin{matrix}6 x \left(x + 2 y - 1\right) & - 6 x^{2} - 12 x y + 6 x - 6 y^{2} + 6 y - 1\\- 6 x^{2} - 12 x y + 6 x - 6 y^{2} + 6 y - 1 & - 12 x y + 6 x + 6 y^{2} - 2\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{11} = \left(\begin{matrix}3 x \left(- x + 6 y - 1\right) & 3 x^{2} - 18 x y + 3 x - 9 y^{2} + 9 y - \frac{3}{2}\\3 x^{2} - 18 x y + 3 x - 9 y^{2} + 9 y - \frac{3}{2} & 3 y \left(- 2 x + 3 y - 1\right)\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{12} = \left(\begin{matrix}12 x \left(- 3 x - 4 y + 3\right) & 12 x^{2} + 24 x y - 12 x + 12 y^{2} - 12 y + 2\\12 x^{2} + 24 x y - 12 x + 12 y^{2} - 12 y + 2 & 12 y \left(- 2 x - y + 1\right)\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{13} = \left(\begin{matrix}12 x \left(- x - 2 y + 1\right) & - 12 x^{2} + 12 x - 12 y^{2} + 12 y - 2\\- 12 x^{2} + 12 x - 12 y^{2} + 12 y - 2 & 12 y \left(- 2 x - y + 1\right)\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{14} = \left(\begin{matrix}12 x \left(x + 2 y - 1\right) & - 12 x^{2} - 24 x y + 12 x - 12 y^{2} + 12 y - 2\\- 12 x^{2} - 24 x y + 12 x - 12 y^{2} + 12 y - 2 & 12 y \left(1 - y\right)\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.

## References

• Arnold, Douglas N. and Winther, Ragnar. Nonconforming mixed elements for elasticity, Numerische Mathematik 13(3), 295–307, 2003. [DOI: 10.1142/S0218202503002507] [BibTeX]

## DefElement stats

 Element added 02 August 2021 Element last updated 10 February 2022