an encyclopedia of finite element definitions

# Arnold–Boffi–Falk

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 Abbreviated names ABF Orders $$0\leqslant k$$ Reference elements quadrilateral Polynomial set $$\mathcal{Z}^{(15)}_{k} \oplus \mathcal{Z}^{(16)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each edge: normal integral moments with an order $$k$$ Lagrange space On each face: integral moments with an order $$k$$ Nédélec (first kind) space,integral moments of the divergence with $$x^{k+1}y^q$$ for q=0,1,...,k, and integral moments of the divergence with $$x^qy^{k+1}$$ for q=0,1,...,k Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "ABF"↓ Show Symfem examples ↓

## Examples

quadrilateral
order 0
quadrilateral
order 1
quadrilateral
order 2
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 3 x \left(x - 1\right)\\\displaystyle 3 y^{2} - 4 y + 1\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 3 x^{2} + 4 x - 1\\\displaystyle 3 y \left(1 - y\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x \left(3 x - 4\right)\\\displaystyle 3 y \left(y - 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 x \left(1 - x\right)\\\displaystyle y \left(4 - 3 y\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 6 x \left(x - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{1})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{3}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{15}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 10 x \left(6 x^{2} y - 4 x^{2} - 9 x y + 6 x + 3 y - 2\right)\\\displaystyle 60 x y^{3} - 108 x y^{2} + 54 x y - 6 x - 40 y^{3} + 72 y^{2} - 36 y + 4\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 50 x \left(- 6 x^{2} y + 4 x^{2} + 9 x y - 6 x - 3 y + 2\right)\\\displaystyle - 60 x y^{3} + 108 x y^{2} - 54 x y + 6 x + 20 y^{3} - 36 y^{2} + 18 y - 2\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 60 x^{3} y + 40 x^{3} + 108 x^{2} y - 72 x^{2} - 54 x y + 36 x + 6 y - 4\\\displaystyle 10 y \left(- 6 x y^{2} + 9 x y - 3 x + 4 y^{2} - 6 y + 2\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 60 x^{3} y - 20 x^{3} - 108 x^{2} y + 36 x^{2} + 54 x y - 18 x - 6 y + 2\\\displaystyle 50 y \left(6 x y^{2} - 9 x y + 3 x - 4 y^{2} + 6 y - 2\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 x \left(- 150 x^{2} y + 100 x^{2} + 234 x y - 156 x - 81 y + 54\right)\\\displaystyle 10 y \left(- 6 x y^{2} + 9 x y - 3 x + 2 y^{2} - 3 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 2 x \left(150 x^{2} y - 50 x^{2} - 234 x y + 78 x + 81 y - 27\right)\\\displaystyle 50 y \left(6 x y^{2} - 9 x y + 3 x - 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3th edge;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 10 x \left(6 x^{2} y - 2 x^{2} - 9 x y + 3 x + 3 y - 1\right)\\\displaystyle 2 y \left(150 x y^{2} - 234 x y + 81 x - 100 y^{2} + 156 y - 54\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3th edge;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 50 x \left(- 6 x^{2} y + 2 x^{2} + 9 x y - 3 x - 3 y + 1\right)\\\displaystyle 2 y \left(- 150 x y^{2} + 234 x y - 81 x + 50 y^{2} - 78 y + 27\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1 - s_{1}\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 24 x \left(- 15 x^{2} y + 10 x^{2} + 24 x y - 16 x - 9 y + 6\right)\\\displaystyle 30 y \left(- 4 x y^{2} + 6 x y - 2 x + 2 y^{2} - 3 y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1 - s_{0}\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 30 x \left(- 4 x^{2} y + 2 x^{2} + 6 x y - 3 x - 2 y + 1\right)\\\displaystyle 24 y \left(- 15 x y^{2} + 24 x y - 9 x + 10 y^{2} - 16 y + 6\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{0}\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 150 x \left(4 x^{2} y - 2 x^{2} - 6 x y + 3 x + 2 y - 1\right)\\\displaystyle 24 y \left(15 x y^{2} - 24 x y + 9 x - 5 y^{2} + 8 y - 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{1}\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 24 x \left(15 x^{2} y - 5 x^{2} - 24 x y + 8 x + 9 y - 3\right)\\\displaystyle 150 y \left(4 x y^{2} - 6 x y + 2 x - 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0}^{2})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 60 x \left(- 6 x^{2} y + 4 x^{2} + 9 x y - 6 x - 3 y + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{1}^{2})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 60 y \left(- 6 x y^{2} + 9 x y - 3 x + 4 y^{2} - 6 y + 2\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0}^{2} s_{1})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 180 x \left(4 x^{2} y - 2 x^{2} - 6 x y + 3 x + 2 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0} s_{1}^{2})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 180 y \left(4 x y^{2} - 6 x y + 2 x - 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3} y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{4}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{4} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{4} y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{4}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{4}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{4}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{29}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 21 x \left(50 x^{3} y^{2} - 60 x^{3} y + 15 x^{3} - 100 x^{2} y^{2} + 120 x^{2} y - 30 x^{2} + 60 x y^{2} - 72 x y + 18 x - 10 y^{2} + 12 y - 3\right)\\\displaystyle 1050 x^{2} y^{4} - 2400 x^{2} y^{3} + 1800 x^{2} y^{2} - 480 x^{2} y + 30 x^{2} - 1260 x y^{4} + 2880 x y^{3} - 2160 x y^{2} + 576 x y - 36 x + 315 y^{4} - 720 y^{3} + 540 y^{2} - 144 y + 9\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 399 x \left(50 x^{3} y^{2} - 60 x^{3} y + 15 x^{3} - 100 x^{2} y^{2} + 120 x^{2} y - 30 x^{2} + 60 x y^{2} - 72 x y + 18 x - 10 y^{2} + 12 y - 3\right)\\\displaystyle 1050 x^{2} y^{4} - 2400 x^{2} y^{3} + 1800 x^{2} y^{2} - 480 x^{2} y + 30 x^{2} - 840 x y^{4} + 1920 x y^{3} - 1440 x y^{2} + 384 x y - 24 x + 105 y^{4} - 240 y^{3} + 180 y^{2} - 48 y + 3\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{105 x \left(50 x^{3} y^{2} - 60 x^{3} y + 15 x^{3} - 100 x^{2} y^{2} + 120 x^{2} y - 30 x^{2} + 60 x y^{2} - 72 x y + 18 x - 10 y^{2} + 12 y - 3\right)}{2}\\\displaystyle - 525 x^{2} y^{4} + 1200 x^{2} y^{3} - 900 x^{2} y^{2} + 240 x^{2} y - 15 x^{2} + 525 x y^{4} - 1200 x y^{3} + 900 x y^{2} - 240 x y + 15 x - \frac{105 y^{4}}{2} + 120 y^{3} - 90 y^{2} + 24 y - \frac{3}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 1050 x^{4} y^{2} + 1260 x^{4} y - 315 x^{4} + 2400 x^{3} y^{2} - 2880 x^{3} y + 720 x^{3} - 1800 x^{2} y^{2} + 2160 x^{2} y - 540 x^{2} + 480 x y^{2} - 576 x y + 144 x - 30 y^{2} + 36 y - 9\\\displaystyle 21 y \left(- 50 x^{2} y^{3} + 100 x^{2} y^{2} - 60 x^{2} y + 10 x^{2} + 60 x y^{3} - 120 x y^{2} + 72 x y - 12 x - 15 y^{3} + 30 y^{2} - 18 y + 3\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 1050 x^{4} y^{2} + 840 x^{4} y - 105 x^{4} + 2400 x^{3} y^{2} - 1920 x^{3} y + 240 x^{3} - 1800 x^{2} y^{2} + 1440 x^{2} y - 180 x^{2} + 480 x y^{2} - 384 x y + 48 x - 30 y^{2} + 24 y - 3\\\displaystyle 399 y \left(- 50 x^{2} y^{3} + 100 x^{2} y^{2} - 60 x^{2} y + 10 x^{2} + 60 x y^{3} - 120 x y^{2} + 72 x y - 12 x - 15 y^{3} + 30 y^{2} - 18 y + 3\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 525 x^{4} y^{2} - 525 x^{4} y + \frac{105 x^{4}}{2} - 1200 x^{3} y^{2} + 1200 x^{3} y - 120 x^{3} + 900 x^{2} y^{2} - 900 x^{2} y + 90 x^{2} - 240 x y^{2} + 240 x y - 24 x + 15 y^{2} - 15 y + \frac{3}{2}\\\displaystyle \frac{105 y \left(- 50 x^{2} y^{3} + 100 x^{2} y^{2} - 60 x^{2} y + 10 x^{2} + 60 x y^{3} - 120 x y^{2} + 72 x y - 12 x - 15 y^{3} + 30 y^{2} - 18 y + 3\right)}{2}\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x \left(6650 x^{3} y^{2} - 7980 x^{3} y + 1995 x^{3} - 13400 x^{2} y^{2} + 16080 x^{2} y - 4020 x^{2} + 8100 x y^{2} - 9720 x y + 2430 x - 1360 y^{2} + 1632 y - 408\right)\\\displaystyle 21 y \left(50 x^{2} y^{3} - 100 x^{2} y^{2} + 60 x^{2} y - 10 x^{2} - 40 x y^{3} + 80 x y^{2} - 48 x y + 8 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 x \left(6650 x^{3} y^{2} - 5320 x^{3} y + 665 x^{3} - 13400 x^{2} y^{2} + 10720 x^{2} y - 1340 x^{2} + 8100 x y^{2} - 6480 x y + 810 x - 1360 y^{2} + 1088 y - 136\right)\\\displaystyle 399 y \left(50 x^{2} y^{3} - 100 x^{2} y^{2} + 60 x^{2} y - 10 x^{2} - 40 x y^{3} + 80 x y^{2} - 48 x y + 8 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{3 x \left(- 6650 x^{3} y^{2} + 6650 x^{3} y - 665 x^{3} + 13400 x^{2} y^{2} - 13400 x^{2} y + 1340 x^{2} - 8100 x y^{2} + 8100 x y - 810 x + 1360 y^{2} - 1360 y + 136\right)}{2}\\\displaystyle \frac{105 y \left(50 x^{2} y^{3} - 100 x^{2} y^{2} + 60 x^{2} y - 10 x^{2} - 40 x y^{3} + 80 x y^{2} - 48 x y + 8 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3th edge;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 21 x \left(- 50 x^{3} y^{2} + 40 x^{3} y - 5 x^{3} + 100 x^{2} y^{2} - 80 x^{2} y + 10 x^{2} - 60 x y^{2} + 48 x y - 6 x + 10 y^{2} - 8 y + 1\right)\\\displaystyle 3 y \left(- 6650 x^{2} y^{3} + 13400 x^{2} y^{2} - 8100 x^{2} y + 1360 x^{2} + 7980 x y^{3} - 16080 x y^{2} + 9720 x y - 1632 x - 1995 y^{3} + 4020 y^{2} - 2430 y + 408\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3th edge;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 399 x \left(- 50 x^{3} y^{2} + 40 x^{3} y - 5 x^{3} + 100 x^{2} y^{2} - 80 x^{2} y + 10 x^{2} - 60 x y^{2} + 48 x y - 6 x + 10 y^{2} - 8 y + 1\right)\\\displaystyle 3 y \left(- 6650 x^{2} y^{3} + 13400 x^{2} y^{2} - 8100 x^{2} y + 1360 x^{2} + 5320 x y^{3} - 10720 x y^{2} + 6480 x y - 1088 x - 665 y^{3} + 1340 y^{2} - 810 y + 136\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3th edge;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{105 x \left(- 50 x^{3} y^{2} + 40 x^{3} y - 5 x^{3} + 100 x^{2} y^{2} - 80 x^{2} y + 10 x^{2} - 60 x y^{2} + 48 x y - 6 x + 10 y^{2} - 8 y + 1\right)}{2}\\\displaystyle \frac{3 y \left(6650 x^{2} y^{3} - 13400 x^{2} y^{2} + 8100 x^{2} y - 1360 x^{2} - 6650 x y^{3} + 13400 x y^{2} - 8100 x y + 1360 x + 665 y^{3} - 1340 y^{2} + 810 y - 136\right)}{2}\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- 18 s_{0} s_{1}^{2} + 24 s_{0} s_{1} - 6 s_{0} + 12 s_{1}^{2} - 16 s_{1} + 4\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 30 x \left(140 x^{3} y^{2} - 168 x^{3} y + 42 x^{3} - 270 x^{2} y^{2} + 324 x^{2} y - 81 x^{2} + 150 x y^{2} - 180 x y + 45 x - 20 y^{2} + 24 y - 6\right)\\\displaystyle 14 y \left(- 75 x^{2} y^{3} + 150 x^{2} y^{2} - 90 x^{2} y + 15 x^{2} + 90 x y^{3} - 180 x y^{2} + 108 x y - 18 x - 20 y^{3} + 40 y^{2} - 24 y + 4\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}18 s_{0} s_{1}^{2} - 24 s_{0} s_{1} + 6 s_{0} - 6 s_{1}^{2} + 8 s_{1} - 2\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 30 x \left(490 x^{3} y^{2} - 588 x^{3} y + 147 x^{3} - 990 x^{2} y^{2} + 1188 x^{2} y - 297 x^{2} + 600 x y^{2} - 720 x y + 180 x - 100 y^{2} + 120 y - 30\right)\\\displaystyle 14 y \left(75 x^{2} y^{3} - 150 x^{2} y^{2} + 90 x^{2} y - 15 x^{2} - 60 x y^{3} + 120 x y^{2} - 72 x y + 12 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- 18 s_{0}^{2} s_{1} + 12 s_{0}^{2} + 24 s_{0} s_{1} - 16 s_{0} - 6 s_{1} + 4\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 14 x \left(- 75 x^{3} y^{2} + 90 x^{3} y - 20 x^{3} + 150 x^{2} y^{2} - 180 x^{2} y + 40 x^{2} - 90 x y^{2} + 108 x y - 24 x + 15 y^{2} - 18 y + 4\right)\\\displaystyle 30 y \left(140 x^{2} y^{3} - 270 x^{2} y^{2} + 150 x^{2} y - 20 x^{2} - 168 x y^{3} + 324 x y^{2} - 180 x y + 24 x + 42 y^{3} - 81 y^{2} + 45 y - 6\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\18 s_{0}^{2} s_{1} - 6 s_{0}^{2} - 24 s_{0} s_{1} + 8 s_{0} + 6 s_{1} - 2\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 14 x \left(75 x^{3} y^{2} - 60 x^{3} y + 5 x^{3} - 150 x^{2} y^{2} + 120 x^{2} y - 10 x^{2} + 90 x y^{2} - 72 x y + 6 x - 15 y^{2} + 12 y - 1\right)\\\displaystyle 30 y \left(490 x^{2} y^{3} - 990 x^{2} y^{2} + 600 x^{2} y - 100 x^{2} - 588 x y^{3} + 1188 x y^{2} - 720 x y + 120 x + 147 y^{3} - 297 y^{2} + 180 y - 30\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\2 s_{0} \left(- 9 s_{0} s_{1} + 6 s_{0} + 6 s_{1} - 4\right)\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 266 x \left(- 75 x^{3} y^{2} + 90 x^{3} y - 20 x^{3} + 150 x^{2} y^{2} - 180 x^{2} y + 40 x^{2} - 90 x y^{2} + 108 x y - 24 x + 15 y^{2} - 18 y + 4\right)\\\displaystyle 30 y \left(140 x^{2} y^{3} - 270 x^{2} y^{2} + 150 x^{2} y - 20 x^{2} - 112 x y^{3} + 216 x y^{2} - 120 x y + 16 x + 14 y^{3} - 27 y^{2} + 15 y - 2\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\2 s_{0} \left(9 s_{0} s_{1} - 3 s_{0} - 6 s_{1} + 2\right)\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 266 x \left(75 x^{3} y^{2} - 60 x^{3} y + 5 x^{3} - 150 x^{2} y^{2} + 120 x^{2} y - 10 x^{2} + 90 x y^{2} - 72 x y + 6 x - 15 y^{2} + 12 y - 1\right)\\\displaystyle 30 y \left(490 x^{2} y^{3} - 990 x^{2} y^{2} + 600 x^{2} y - 100 x^{2} - 392 x y^{3} + 792 x y^{2} - 480 x y + 80 x + 49 y^{3} - 99 y^{2} + 60 y - 10\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}2 s_{1} \left(- 9 s_{0} s_{1} + 6 s_{0} + 6 s_{1} - 4\right)\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 30 x \left(140 x^{3} y^{2} - 112 x^{3} y + 14 x^{3} - 270 x^{2} y^{2} + 216 x^{2} y - 27 x^{2} + 150 x y^{2} - 120 x y + 15 x - 20 y^{2} + 16 y - 2\right)\\\displaystyle 266 y \left(- 75 x^{2} y^{3} + 150 x^{2} y^{2} - 90 x^{2} y + 15 x^{2} + 90 x y^{3} - 180 x y^{2} + 108 x y - 18 x - 20 y^{3} + 40 y^{2} - 24 y + 4\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}2 s_{1} \left(9 s_{0} s_{1} - 6 s_{0} - 3 s_{1} + 2\right)\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 30 x \left(490 x^{3} y^{2} - 392 x^{3} y + 49 x^{3} - 990 x^{2} y^{2} + 792 x^{2} y - 99 x^{2} + 600 x y^{2} - 480 x y + 60 x - 100 y^{2} + 80 y - 10\right)\\\displaystyle 266 y \left(75 x^{2} y^{3} - 150 x^{2} y^{2} + 90 x^{2} y - 15 x^{2} - 60 x y^{3} + 120 x y^{2} - 72 x y + 12 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\12 s_{0} \left(3 s_{0} s_{1} - 2 s_{0} - 3 s_{1} + 2\right)\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 70 x \left(- 75 x^{3} y^{2} + 90 x^{3} y - 20 x^{3} + 150 x^{2} y^{2} - 180 x^{2} y + 40 x^{2} - 90 x y^{2} + 108 x y - 24 x + 15 y^{2} - 18 y + 4\right)\\\displaystyle 10 y \left(14 y^{3} - 27 y^{2} + 15 y - 2\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}12 s_{1} \left(- 3 s_{0} s_{1} + 3 s_{0} + 2 s_{1} - 2\right)\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 10 x \left(- 14 x^{3} + 27 x^{2} - 15 x + 2\right)\\\displaystyle 70 y \left(75 x^{2} y^{3} - 150 x^{2} y^{2} + 90 x^{2} y - 15 x^{2} - 90 x y^{3} + 180 x y^{2} - 108 x y + 18 x + 20 y^{3} - 40 y^{2} + 24 y - 4\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}12 s_{1} \left(3 s_{0} s_{1} - 3 s_{0} - s_{1} + 1\right)\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 10 x \left(- 49 x^{3} + 99 x^{2} - 60 x + 10\right)\\\displaystyle 70 y \left(- 75 x^{2} y^{3} + 150 x^{2} y^{2} - 90 x^{2} y + 15 x^{2} + 60 x y^{3} - 120 x y^{2} + 72 x y - 12 x - 5 y^{3} + 10 y^{2} - 6 y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\12 s_{0} \left(- 3 s_{0} s_{1} + s_{0} + 3 s_{1} - 1\right)\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 70 x \left(75 x^{3} y^{2} - 60 x^{3} y + 5 x^{3} - 150 x^{2} y^{2} + 120 x^{2} y - 10 x^{2} + 90 x y^{2} - 72 x y + 6 x - 15 y^{2} + 12 y - 1\right)\\\displaystyle 10 y \left(49 y^{3} - 99 y^{2} + 60 y - 10\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0}^{3})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 420 x \left(50 x^{3} y^{2} - 60 x^{3} y + 15 x^{3} - 100 x^{2} y^{2} + 120 x^{2} y - 30 x^{2} + 60 x y^{2} - 72 x y + 18 x - 10 y^{2} + 12 y - 3\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{1}^{3})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 420 y \left(50 x^{2} y^{3} - 100 x^{2} y^{2} + 60 x^{2} y - 10 x^{2} - 60 x y^{3} + 120 x y^{2} - 72 x y + 12 x + 15 y^{3} - 30 y^{2} + 18 y - 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0}^{3} s_{1})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 1680 x \left(- 75 x^{3} y^{2} + 80 x^{3} y - 15 x^{3} + 150 x^{2} y^{2} - 160 x^{2} y + 30 x^{2} - 90 x y^{2} + 96 x y - 18 x + 15 y^{2} - 16 y + 3\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{27}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0} s_{1}^{3})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 1680 y \left(- 75 x^{2} y^{3} + 150 x^{2} y^{2} - 90 x^{2} y + 15 x^{2} + 80 x y^{3} - 160 x y^{2} + 96 x y - 16 x - 15 y^{3} + 30 y^{2} - 18 y + 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{28}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0}^{3} s_{1}^{2})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 4200 x \left(30 x^{3} y^{2} - 30 x^{3} y + 5 x^{3} - 60 x^{2} y^{2} + 60 x^{2} y - 10 x^{2} + 36 x y^{2} - 36 x y + 6 x - 6 y^{2} + 6 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{29}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0}^{2} s_{1}^{3})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4200 y \left(30 x^{2} y^{3} - 60 x^{2} y^{2} + 36 x^{2} y - 6 x^{2} - 30 x y^{3} + 60 x y^{2} - 36 x y + 6 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\end{array}\right)$$

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

## References

• Arnold, D. N., Boffi, D., and Falk, R. S. Quadrilateral H(div) Finite Elements, SIAM Journal on Numerical Analysis 42(5), 2429–2451, 2005. [DOI: 10.1137/S0036142903431924] [BibTeX]

## DefElement stats

 Element added 01 December 2021 Element last updated 01 December 2021