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# Degree 2 Nédélec (first kind) on a tetrahedron

◀ Back to Nédélec (first kind) definition page In this example:
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle - x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle - x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle - x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle - y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle - y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle - x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle - x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle - x y\\\displaystyle 0\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{19}\}$$
• 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 $$e_{0}$$.

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

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 $$e_{0}$$.

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

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 $$e_{1}$$.

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

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 $$e_{1}$$.

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

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 $$e_{2}$$.

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

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 $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 4 y \left(1 - 2 y\right)\\\displaystyle 2 x \left(4 y - 1\right)\\\displaystyle 0\end{array}\right)$$

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 $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 16 x y + 8 x z - 12 x + 8 y^{2} + 8 y z - 12 y - 6 z + 4\end{array}\right)$$

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 $$e_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle - 8 x z + 2 x - 8 y z + 2 y + 6 z - 2\end{array}\right)$$

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 $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 8 x y + 16 x z - 12 x + 8 y z - 6 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)$$

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 $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x - 8 y z + 6 y + 2 z - 2\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)$$

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 $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 8 x y + 8 x z - 6 x + 8 y^{2} + 16 y z - 12 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)$$

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 $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle - 8 x y - 8 x z + 6 x + 2 y + 2 z - 2\\\displaystyle 4 x \left(2 x - 1\right)\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element. $$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}- \frac{\sqrt{3}}{3}\\\frac{\sqrt{3}}{3}\\0\end{array}\right)$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle 16 x z\\\displaystyle - 8 x y\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}- \frac{\sqrt{3}}{3}\\0\\\frac{\sqrt{3}}{3}\end{array}\right)$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle - 8 x z\\\displaystyle 16 x y\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 y \left(x + y + 2 z - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element. $$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$
where $$f_{1}$$ is the 1st face.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(x + 2 y + z - 1\right)\\\displaystyle 8 y \left(- 2 x - 2 y - z + 2\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element. $$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 8 z \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(x + y + 2 z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element. $$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 8 z \left(2 x + y + z - 1\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(- 2 x - 2 y - z + 2\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element. $$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$
where $$f_{3}$$ is the 3rd face.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 8 y \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x \left(x + 2 y + z - 1\right)\\\displaystyle 8 x y\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)$$
where $$f_{3}$$ is the 3rd face.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 8 y \left(2 x + y + z - 1\right)\\\displaystyle 8 x \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 x y\end{array}\right)$$

This DOF is associated with face 3 of the reference element.