Degree 2 Tiniest tensor H(curl) on a quadrilateral

• $R$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:

• $\mathcal{V}$ is spanned by: $\left(\begin{array}{c}{\displaystyle 1}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 1}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle y}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle y}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle y^2}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle y^2}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle x}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle x}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle xy}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle xy}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle x{y}^2}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle x{y}^2}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle x^2}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle x^2}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle x^{2}y}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle x^{2}y}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle x^2{y}^2}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle x^2{y}^2}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 2y(2y^2-3y+1)}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 2x(-2x^2+3x-1)}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 2y(12x^2{y}^2-18x^{2}y+6x^2-12x{y}^2+18xy-6x+2y^2-3y+1)}\\ {\displaystyle 2x(-12x^2{y}^2+12x^{2}y-2x^2+18x{y}^2-18xy+3x-6y^2+6y-1)}\end{array}\right)$
• $\mathcal{L}=\{l_0,...,l_{20}\}$
• Functionals and basis functions:

${\displaystyle l_0:\mathit{v}\mapsto {\displaystyle {\int }_{e_0}\mathit{v}\cdot (2s_0^2-3s_0+1){\hat{\mathit{t}}}_0}}$
where $e_0$ is the 0th edge;
${\hat{\mathit{t}}}_0$ is the tangent to edge 0;
and $s_0,s_1$ is a parametrisation of $e_0$.

${\displaystyle {\mathit{\varphi }}_0=\left(\begin{array}{c}{\displaystyle -150x^2{y}^3+315x^2{y}^2-195x^{2}y+30x^2+150x{y}^3-\frac{663x{y}^2}{2}+\frac{435xy}{2}-36x-35y^3+\frac{315y^2}{4}-\frac{211y}{4}+9}\\ {\displaystyle \frac{x(600x^2{y}^2-600x^{2}y+100x^2-1260x{y}^2+1266xy-213x+660y^2-666y+113)}{4}}\end{array}\right)}$

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

${\displaystyle l_1:\mathit{v}\mapsto {\displaystyle {\int }_{e_0}\mathit{v}\cdot (s_0(2s_0-1)){\hat{\mathit{t}}}_0}}$
where $e_0$ is the 0th edge;
${\hat{\mathit{t}}}_0$ is the tangent to edge 0;
and $s_0,s_1$ is a parametrisation of $e_0$.

${\displaystyle {\mathit{\varphi }}_1=\left(\begin{array}{c}{\displaystyle -150x^2{y}^3+315x^2{y}^2-195x^{2}y+30x^2+150x{y}^3-\frac{597x{y}^2}{2}+\frac{345xy}{2}-24x-35y^3+\frac{249y^2}{4}-\frac{121y}{4}+3}\\ {\displaystyle \frac{x(600x^2{y}^2-600x^{2}y+100x^2-540x{y}^2+534xy-87x-60y^2+66y-13)}{4}}\end{array}\right)}$

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

${\displaystyle l_2:\mathit{v}\mapsto {\displaystyle {\int }_{e_0}\mathit{v}\cdot (4s_0(1-s_0)){\hat{\mathit{t}}}_0}}$
where $e_0$ is the 0th edge;
${\hat{\mathit{t}}}_0$ is the tangent to edge 0;
and $s_0,s_1$ is a parametrisation of $e_0$.

${\displaystyle {\mathit{\varphi }}_2=\left(\begin{array}{c}{\displaystyle 75x^2{y}^3-\frac{315x^2{y}^2}{2}+\frac{195x^{2}y}{2}-15x^2-75x{y}^3+\frac{315x{y}^2}{2}-\frac{195xy}{2}+15x+\frac{5y^3}{2}-\frac{33y^2}{4}+\frac{29y}{4}-\frac{3}{2}}\\ {\displaystyle \frac{25x(-12x^2{y}^2+12x^{2}y-2x^2+18x{y}^2-18xy+3x-6y^2+6y-1)}{4}}\end{array}\right)}$

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

${\displaystyle l_3:\mathit{v}\mapsto {\displaystyle {\int }_{e_1}\mathit{v}\cdot (2s_0^2-3s_0+1){\hat{\mathit{t}}}_1}}$
where $e_1$ is the 1st edge;
${\hat{\mathit{t}}}_1$ is the tangent to edge 1;
and $s_0,s_1$ is a parametrisation of $e_1$.

${\displaystyle {\mathit{\varphi }}_3=\left(\begin{array}{c}{\displaystyle \frac{y(600x^2{y}^2-1260x^{2}y+660x^2-600x{y}^2+1266xy-666x+100y^2-213y+113)}{4}}\\ {\displaystyle -150x^3{y}^2+150x^{3}y-35x^3+315x^2{y}^2-\frac{663x^{2}y}{2}+\frac{315x^2}{4}-195x{y}^2+\frac{435xy}{2}-\frac{211x}{4}+30y^2-36y+9}\end{array}\right)}$

This DOF is associated with edge 1 of the reference element.

${\displaystyle l_4:\mathit{v}\mapsto {\displaystyle {\int }_{e_1}\mathit{v}\cdot (s_0(2s_0-1)){\hat{\mathit{t}}}_1}}$
where $e_1$ is the 1st edge;
${\hat{\mathit{t}}}_1$ is the tangent to edge 1;
and $s_0,s_1$ is a parametrisation of $e_1$.

${\displaystyle {\mathit{\varphi }}_4=\left(\begin{array}{c}{\displaystyle \frac{y(600x^2{y}^2-540x^{2}y-60x^2-600x{y}^2+534xy+66x+100y^2-87y-13)}{4}}\\ {\displaystyle -150x^3{y}^2+150x^{3}y-35x^3+315x^2{y}^2-\frac{597x^{2}y}{2}+\frac{249x^2}{4}-195x{y}^2+\frac{345xy}{2}-\frac{121x}{4}+30y^2-24y+3}\end{array}\right)}$

This DOF is associated with edge 1 of the reference element.

${\displaystyle l_5:\mathit{v}\mapsto {\displaystyle {\int }_{e_1}\mathit{v}\cdot (4s_0(1-s_0)){\hat{\mathit{t}}}_1}}$
where $e_1$ is the 1st edge;
${\hat{\mathit{t}}}_1$ is the tangent to edge 1;
and $s_0,s_1$ is a parametrisation of $e_1$.

${\displaystyle {\mathit{\varphi }}_5=\left(\begin{array}{c}{\displaystyle \frac{25y(-12x^2{y}^2+18x^{2}y-6x^2+12x{y}^2-18xy+6x-2y^2+3y-1)}{4}}\\ {\displaystyle 75x^3{y}^2-75x^{3}y+\frac{5x^3}{2}-\frac{315x^2{y}^2}{2}+\frac{315x^{2}y}{2}-\frac{33x^2}{4}+\frac{195x{y}^2}{2}-\frac{195xy}{2}+\frac{29x}{4}-15y^2+15y-\frac{3}{2}}\end{array}\right)}$

This DOF is associated with edge 1 of the reference element.

${\displaystyle l_6:\mathit{v}\mapsto {\displaystyle {\int }_{e_2}\mathit{v}\cdot (2s_0^2-3s_0+1){\hat{\mathit{t}}}_2}}$
where $e_2$ is the 2nd edge;
${\hat{\mathit{t}}}_2$ is the tangent to edge 2;
and $s_0,s_1$ is a parametrisation of $e_2$.

${\displaystyle {\mathit{\varphi }}_6=\left(\begin{array}{c}{\displaystyle \frac{y(-600x^2{y}^2+1260x^{2}y-660x^2+600x{y}^2-1254xy+654x-100y^2+207y-107)}{4}}\\ {\displaystyle \frac{x(600x^2{y}^2-600x^{2}y+140x^2-540x{y}^2+474xy-105x+60y^2-18y+1)}{4}}\end{array}\right)}$

This DOF is associated with edge 2 of the reference element.

${\displaystyle l_7:\mathit{v}\mapsto {\displaystyle {\int }_{e_2}\mathit{v}\cdot (s_0(2s_0-1)){\hat{\mathit{t}}}_2}}$
where $e_2$ is the 2nd edge;
${\hat{\mathit{t}}}_2$ is the tangent to edge 2;
and $s_0,s_1$ is a parametrisation of $e_2$.

${\displaystyle {\mathit{\varphi }}_7=\left(\begin{array}{c}{\displaystyle \frac{y(-600x^2{y}^2+540x^{2}y+60x^2+600x{y}^2-546xy-54x-100y^2+93y+7)}{4}}\\ {\displaystyle \frac{x(600x^2{y}^2-600x^{2}y+140x^2-540x{y}^2+606xy-171x+60y^2-102y+43)}{4}}\end{array}\right)}$

This DOF is associated with edge 2 of the reference element.

${\displaystyle l_8:\mathit{v}\mapsto {\displaystyle {\int }_{e_2}\mathit{v}\cdot (4s_0(1-s_0)){\hat{\mathit{t}}}_2}}$
where $e_2$ is the 2nd edge;
${\hat{\mathit{t}}}_2$ is the tangent to edge 2;
and $s_0,s_1$ is a parametrisation of $e_2$.

${\displaystyle {\mathit{\varphi }}_8=\left(\begin{array}{c}{\displaystyle \frac{25y(12x^2{y}^2-18x^{2}y+6x^2-12x{y}^2+18xy-6x+2y^2-3y+1)}{4}}\\ {\displaystyle \frac{x(-300x^2{y}^2+300x^{2}y-10x^2+270x{y}^2-270xy-3x-30y^2+30y+7)}{4}}\end{array}\right)}$

This DOF is associated with edge 2 of the reference element.

${\displaystyle l_9:\mathit{v}\mapsto {\displaystyle {\int }_{e_3}\mathit{v}\cdot (2s_0^2-3s_0+1){\hat{\mathit{t}}}_3}}$
where $e_3$ is the 3rd edge;
${\hat{\mathit{t}}}_3$ is the tangent to edge 3;
and $s_0,s_1$ is a parametrisation of $e_3$.

${\displaystyle {\mathit{\varphi }}_9=\left(\begin{array}{c}{\displaystyle \frac{y(600x^2{y}^2-540x^{2}y+60x^2-600x{y}^2+474xy-18x+140y^2-105y+1)}{4}}\\ {\displaystyle \frac{x(-600x^2{y}^2+600x^{2}y-100x^2+1260x{y}^2-1254xy+207x-660y^2+654y-107)}{4}}\end{array}\right)}$

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

${\displaystyle l_{10}:\mathit{v}\mapsto {\displaystyle {\int }_{e_3}\mathit{v}\cdot (s_0(2s_0-1)){\hat{\mathit{t}}}_3}}$
where $e_3$ is the 3rd edge;
${\hat{\mathit{t}}}_3$ is the tangent to edge 3;
and $s_0,s_1$ is a parametrisation of $e_3$.

${\displaystyle {\mathit{\varphi }}_{10}=\left(\begin{array}{c}{\displaystyle \frac{y(600x^2{y}^2-540x^{2}y+60x^2-600x{y}^2+606xy-102x+140y^2-171y+43)}{4}}\\ {\displaystyle \frac{x(-600x^2{y}^2+600x^{2}y-100x^2+540x{y}^2-546xy+93x+60y^2-54y+7)}{4}}\end{array}\right)}$

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

${\displaystyle l_{11}:\mathit{v}\mapsto {\displaystyle {\int }_{e_3}\mathit{v}\cdot (4s_0(1-s_0)){\hat{\mathit{t}}}_3}}$
where $e_3$ is the 3rd edge;
${\hat{\mathit{t}}}_3$ is the tangent to edge 3;
and $s_0,s_1$ is a parametrisation of $e_3$.

${\displaystyle {\mathit{\varphi }}_{11}=\left(\begin{array}{c}{\displaystyle \frac{y(-300x^2{y}^2+270x^{2}y-30x^2+300x{y}^2-270xy+30x-10y^2-3y+7)}{4}}\\ {\displaystyle \frac{25x(12x^2{y}^2-12x^{2}y+2x^2-18x{y}^2+18xy-3x+6y^2-6y+1)}{4}}\end{array}\right)}$

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

${\displaystyle l_{12}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle 1}\\ {\displaystyle 0}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element.

${\displaystyle {\mathit{\varphi }}_{12}=\left(\begin{array}{c}{\displaystyle 6y(150x^2{y}^2-255x^{2}y+105x^2-150x{y}^2+258xy-108x+35y^2-60y+25)}\\ {\displaystyle 6x(-150x^2{y}^2+150x^{2}y-25x^2+315x{y}^2-318xy+54x-165y^2+168y-29)}\end{array}\right)}$

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

${\displaystyle l_{13}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle 2s_1}\\ {\displaystyle 0}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{13}=\left(\begin{array}{c}{\displaystyle 15y(-60x^2{y}^2+90x^{2}y-30x^2+60x{y}^2-90xy+30x-14y^2+21y-7)}\\ {\displaystyle 15x(60x^2{y}^2-60x^{2}y+10x^2-126x{y}^2+126xy-21x+66y^2-66y+11)}\end{array}\right)}$

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

${\displaystyle l_{14}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle -1}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element.

${\displaystyle {\mathit{\varphi }}_{14}=\left(\begin{array}{c}{\displaystyle 6y(150x^2{y}^2-315x^{2}y+165x^2-150x{y}^2+318xy-168x+25y^2-54y+29)}\\ {\displaystyle 6x(-150x^2{y}^2+150x^{2}y-35x^2+255x{y}^2-258xy+60x-105y^2+108y-25)}\end{array}\right)}$

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

${\displaystyle l_{15}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle s_0}\\ {\displaystyle -s_1}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{15}=\left(\begin{array}{c}{\displaystyle 36y(-150x^2{y}^2+255x^{2}y-105x^2+150x{y}^2-256xy+106x-25y^2+43y-18)}\\ {\displaystyle 36x(150x^2{y}^2-150x^{2}y+25x^2-255x{y}^2+256xy-43x+105y^2-106y+18)}\end{array}\right)}$

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

${\displaystyle l_{16}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle 2s_0{s}_1}\\ {\displaystyle -s_1^2}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{16}=\left(\begin{array}{c}{\displaystyle 450y(12x^2{y}^2-18x^{2}y+6x^2-12x{y}^2+18xy-6x+2y^2-3y+1)}\\ {\displaystyle 90x(-60x^2{y}^2+60x^{2}y-10x^2+102x{y}^2-102xy+17x-42y^2+42y-7)}\end{array}\right)}$

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

${\displaystyle l_{17}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle -2s_0}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{17}=\left(\begin{array}{c}{\displaystyle 15y(-60x^2{y}^2+126x^{2}y-66x^2+60x{y}^2-126xy+66x-10y^2+21y-11)}\\ {\displaystyle 15x(60x^2{y}^2-60x^{2}y+14x^2-90x{y}^2+90xy-21x+30y^2-30y+7)}\end{array}\right)}$

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

${\displaystyle l_{18}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle s_0^2}\\ {\displaystyle -2s_0{s}_1}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{18}=\left(\begin{array}{c}{\displaystyle 90y(60x^2{y}^2-102x^{2}y+42x^2-60x{y}^2+102xy-42x+10y^2-17y+7)}\\ {\displaystyle 450x(-12x^2{y}^2+12x^{2}y-2x^2+18x{y}^2-18xy+3x-6y^2+6y-1)}\end{array}\right)}$

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

${\displaystyle l_{19}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle 2s_0^2{s}_1}\\ {\displaystyle -2s_0{s}_1^2}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{19}=\left(\begin{array}{c}{\displaystyle 450y(-12x^2{y}^2+18x^{2}y-6x^2+12x{y}^2-18xy+6x-2y^2+3y-1)}\\ {\displaystyle 450x(12x^2{y}^2-12x^{2}y+2x^2-18x{y}^2+18xy-3x+6y^2-6y+1)}\end{array}\right)}$

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

${\displaystyle l_{20}:\mathbf{v}\mapsto {\displaystyle {\int }_R\left(\begin{array}{c}{\displaystyle s_1(-2s_0{s}_1+2s_0+s_1-1)}\\ {\displaystyle s_0(-2s_0{s}_1+s_0+2s_1-1)}\end{array}\right)\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_{20}=\left(\begin{array}{c}{\displaystyle 45y(-2xy+2x+y-1)}\\ {\displaystyle 45x(-2xy+x+2y-1)}\end{array}\right)}$

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