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# Degree 1 serendipity H(curl) on a hexahedron

◀ Back to serendipity H(curl) definition page In this example:
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle - x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle - x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 2 y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y z\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 2 x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle x z\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 2 x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x z\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle x z^{2}\\\displaystyle 2 x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 2 x y z\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y z\\\displaystyle x^{2} z\\\displaystyle x^{2} y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{23}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{0}$$.

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

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

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

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

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

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

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

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

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

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 3 z \left(y z - y - z + 1\right)\\\displaystyle 3 z \left(x z - x - z + 1\right)\\\displaystyle 6 x y z - 2 x y - 6 x z + 2 x - 6 y z + 2 y + 6 z - 2\end{array}\right)$$

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

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

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3rd edge;
$$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 y \left(- y z + y + z - 1\right)\\\displaystyle 2 x \left(- 3 y z + 3 y + z - 1\right)\\\displaystyle 3 x y \left(1 - y\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
$$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 z \left(y z - y - z + 1\right)\\\displaystyle 3 x z \left(z - 1\right)\\\displaystyle 2 x \left(3 y z - 2 y - 3 z + 2\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element. $$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
$$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 3 z \left(- y z + y + z - 1\right)\\\displaystyle 3 x z \left(1 - z\right)\\\displaystyle 2 x \left(- 3 y z + y + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element. $$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
$$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 2 y \left(3 x z - 3 x - 2 z + 2\right)\\\displaystyle 3 x \left(x z - x - z + 1\right)\\\displaystyle 3 x y \left(x - 1\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element. $$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
$$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 2 y \left(- 3 x z + 3 x + z - 1\right)\\\displaystyle 3 x \left(- x z + x + z - 1\right)\\\displaystyle 3 x y \left(1 - x\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element. $$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
$$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 y z \left(z - 1\right)\\\displaystyle 3 z \left(x z - x - z + 1\right)\\\displaystyle 2 y \left(3 x z - 2 x - 3 z + 2\right)\end{array}\right)$$

This DOF is associated with edge 6 of the reference element. $$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
$$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - z\right)\\\displaystyle 3 z \left(- x z + x + z - 1\right)\\\displaystyle 2 y \left(- 3 x z + x + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with edge 6 of the reference element. $$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
$$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{7}$$.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - z\right)\\\displaystyle 3 x z \left(1 - z\right)\\\displaystyle 2 x y \left(2 - 3 z\right)\end{array}\right)$$

This DOF is associated with edge 7 of the reference element. $$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
$$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{7}$$.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 3 y z \left(z - 1\right)\\\displaystyle 3 x z \left(z - 1\right)\\\displaystyle 2 x y \left(3 z - 1\right)\end{array}\right)$$

This DOF is associated with edge 7 of the reference element. $$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
$$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 2 z \left(3 x y - 3 x - 2 y + 2\right)\\\displaystyle 3 x z \left(x - 1\right)\\\displaystyle 3 x \left(x y - x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 8 of the reference element. $$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
$$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 2 z \left(- 3 x y + 3 x + y - 1\right)\\\displaystyle 3 x z \left(1 - x\right)\\\displaystyle 3 x \left(- x y + x + y - 1\right)\end{array}\right)$$

This DOF is associated with edge 8 of the reference element. $$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
$$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{9}$$.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 3 y z \left(y - 1\right)\\\displaystyle 2 z \left(3 x y - 2 x - 3 y + 2\right)\\\displaystyle 3 y \left(x y - x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 9 of the reference element. $$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
$$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{9}$$.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - y\right)\\\displaystyle 2 z \left(- 3 x y + x + 3 y - 1\right)\\\displaystyle 3 y \left(- x y + x + y - 1\right)\end{array}\right)$$

This DOF is associated with edge 9 of the reference element. $$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
$$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{10}$$.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - y\right)\\\displaystyle 2 x z \left(2 - 3 y\right)\\\displaystyle 3 x y \left(1 - y\right)\end{array}\right)$$

This DOF is associated with edge 10 of the reference element. $$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
$$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{10}$$.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 3 y z \left(y - 1\right)\\\displaystyle 2 x z \left(3 y - 1\right)\\\displaystyle 3 x y \left(y - 1\right)\end{array}\right)$$

This DOF is associated with edge 10 of the reference element. $$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
$$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{11}$$.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 2 y z \left(2 - 3 x\right)\\\displaystyle 3 x z \left(1 - x\right)\\\displaystyle 3 x y \left(1 - x\right)\end{array}\right)$$

This DOF is associated with edge 11 of the reference element. $$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
$$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$e_{11}$$.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 2 y z \left(3 x - 1\right)\\\displaystyle 3 x z \left(x - 1\right)\\\displaystyle 3 x y \left(x - 1\right)\end{array}\right)$$

This DOF is associated with edge 11 of the reference element.