an encyclopedia of finite element definitions

# Degree 3 Huang–Zhang on a quadrilateral

◀ Back to Huang–Zhang definition page In this example:
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3} y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{3}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{23}\}$$
• 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;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 300 x^{2} y^{3} + 540 x^{2} y^{2} - 270 x^{2} y + 30 x^{2} + 360 x y^{3} - 648 x y^{2} + 324 x y - 36 x - 90 y^{3} + 162 y^{2} - 81 y + 9\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 300 x^{2} y^{3} + 540 x^{2} y^{2} - 270 x^{2} y + 30 x^{2} + 240 x y^{3} - 432 x y^{2} + 216 x y - 24 x - 30 y^{3} + 54 y^{2} - 27 y + 3\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 150 x^{2} y^{3} - 270 x^{2} y^{2} + 135 x^{2} y - 15 x^{2} - 150 x y^{3} + 270 x y^{2} - 135 x y + 15 x + 15 y^{3} - 27 y^{2} + \frac{27 y}{2} - \frac{3}{2}\end{array}\right)$$

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;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 300 x^{3} y^{2} - 360 x^{3} y + 90 x^{3} - 540 x^{2} y^{2} + 648 x^{2} y - 162 x^{2} + 270 x y^{2} - 324 x y + 81 x - 30 y^{2} + 36 y - 9\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 300 x^{3} y^{2} - 240 x^{3} y + 30 x^{3} - 540 x^{2} y^{2} + 432 x^{2} y - 54 x^{2} + 270 x y^{2} - 216 x y + 27 x - 30 y^{2} + 24 y - 3\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 150 x^{3} y^{2} + 150 x^{3} y - 15 x^{3} + 270 x^{2} y^{2} - 270 x^{2} y + 27 x^{2} - 135 x y^{2} + 135 x y - \frac{27 x}{2} + 15 y^{2} - 15 y + \frac{3}{2}\\\displaystyle 0\end{array}\right)$$

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;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x \left(- 100 x^{2} y^{2} + 120 x^{2} y - 30 x^{2} + 120 x y^{2} - 144 x y + 36 x - 30 y^{2} + 36 y - 9\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 x \left(- 100 x^{2} y^{2} + 80 x^{2} y - 10 x^{2} + 120 x y^{2} - 96 x y + 12 x - 30 y^{2} + 24 y - 3\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{3 x \left(100 x^{2} y^{2} - 100 x^{2} y + 10 x^{2} - 120 x y^{2} + 120 x y - 12 x + 30 y^{2} - 30 y + 3\right)}{2}\\\displaystyle 0\end{array}\right)$$

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;
and $$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 y \left(100 x^{2} y^{2} - 120 x^{2} y + 30 x^{2} - 120 x y^{2} + 144 x y - 36 x + 30 y^{2} - 36 y + 9\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 y \left(100 x^{2} y^{2} - 120 x^{2} y + 30 x^{2} - 80 x y^{2} + 96 x y - 24 x + 10 y^{2} - 12 y + 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 y \left(- 100 x^{2} y^{2} + 120 x^{2} y - 30 x^{2} + 100 x y^{2} - 120 x y + 30 x - 10 y^{2} + 12 y - 3\right)}{2}\end{array}\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 12 x \left(15 x^{2} y^{2} - 20 x^{2} y + 5 x^{2} - 24 x y^{2} + 32 x y - 8 x + 9 y^{2} - 12 y + 3\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 12 x \left(- 15 x^{2} y^{2} + 20 x^{2} y - 5 x^{2} + 21 x y^{2} - 28 x y + 7 x - 6 y^{2} + 8 y - 2\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 y \left(15 x^{2} y^{2} - 24 x^{2} y + 9 x^{2} - 20 x y^{2} + 32 x y - 12 x + 5 y^{2} - 8 y + 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 y \left(- 15 x^{2} y^{2} + 21 x^{2} y - 6 x^{2} + 20 x y^{2} - 28 x y + 8 x - 5 y^{2} + 7 y - 2\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x y \left(15 x y^{2} - 24 x y + 9 x - 10 y^{2} + 16 y - 6\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x y \left(- 15 x y^{2} + 21 x y - 6 x + 10 y^{2} - 14 y + 4\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 12 x y \left(15 x^{2} y - 10 x^{2} - 24 x y + 16 x + 9 y - 6\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 12 x y \left(- 15 x^{2} y + 10 x^{2} + 21 x y - 14 x - 6 y + 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{20}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 72 x y \left(- 5 x^{2} y + 5 x^{2} + 8 x y - 8 x - 3 y + 3\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{21}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 72 x y \left(- 5 x y^{2} + 8 x y - 3 x + 5 y^{2} - 8 y + 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{22}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 360 x y \left(2 x^{2} y - 2 x^{2} - 3 x y + 3 x + y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{23}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{1}\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 360 x y \left(2 x y^{2} - 3 x y + x - 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

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