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# Degree 1 Nédélec (first kind) on a hexahedron

◀ Back to Nédélec (first kind) 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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(1 - x\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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