an encyclopedia of finite element definitions

# Tiniest tensor

 Alternative names TNT Cockburn–fu names $$\left[S_{3,k}^\square\right]_{0}$$ Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{Q}_{k} \oplus \mathcal{Z}^{(49)}_{k}$$ (quadrilateral) $$\mathcal{Q}_{k} \oplus \mathcal{Z}^{(50)}_{k} \oplus \mathcal{Z}^{(51)}_{k} \oplus \mathcal{Z}^{(52)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: integral moments with $$\frac{\partial}{\partial x}f$$ for each $$f$$ in an order $$k$$ Lagrange space On each face: integral moments with $$\Delta f$$ for each $$f$$ in an order $$k$$ Lagrange space such that the trace of $$f$$ on the edges of the face is 0 On each volume: integral moments of gradient with $$\nabla f$$ for each $$f$$ in an order $$k$$ Lagrange space such that the trace of $$f$$ on the faces of the volume is 0 Number of DOFs quadrilateral: $$(k+1)^2 + 4$$hexahedron: $$(k+1)^3 + 12$$ Categories Scalar-valued elements

## Implementations

 Symfem "TNT" (quadrilateral, hexahedron)↓ Show Symfem examples ↓

## Examples

order 1
order 2
order 3
hexahedron
order 1
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$y$$, $$x$$, $$x y$$, $$\frac{3 x \left(- x y + x + y - 1\right)}{2}$$, $$\frac{3 x y \left(x - 1\right)}{2}$$, $$\frac{3 y \left(- x y + x + y - 1\right)}{2}$$, $$\frac{3 x y \left(y - 1\right)}{2}$$
• $$\mathcal{L}=\{l_0,...,l_{7}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 7 x y - 4 x + 3 y^{2} - 4 y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{1} = x \left(- 3 x y + 3 x + 3 y^{2} - y - 2\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{2} = y \left(3 x^{2} - 3 x y - x + 3 y - 2\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,1)$$

$$\displaystyle \phi_{3} = x y \left(3 x + 3 y - 5\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{4} = 6 x \left(x y - x - y + 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{5} = 6 y \left(x y - x - y + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{6} = 6 x y \left(1 - y\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{7} = 6 x y \left(1 - x\right)$$

This DOF is associated with edge 3 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$y$$, $$y^{2}$$, $$x$$, $$x y$$, $$x y^{2}$$, $$x^{2}$$, $$x^{2} y$$, $$x^{2} y^{2}$$, $$2 x \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)$$, $$2 x y \left(2 x^{2} - 3 x + 1\right)$$, $$2 y \left(- 2 x y^{2} + 3 x y - x + 2 y^{2} - 3 y + 1\right)$$, $$2 x y \left(2 y^{2} - 3 y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{12}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 10 x^{3} y - 10 x^{3} + 9 x^{2} y^{2} - 27 x^{2} y + 18 x^{2} + 10 x y^{3} - 27 x y^{2} + 26 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{1} = x \left(- 10 x^{2} y + 10 x^{2} + 9 x y^{2} + 3 x y - 12 x - 10 y^{3} + 9 y^{2} - 2 y + 3\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{2} = y \left(- 10 x^{3} + 9 x^{2} y + 9 x^{2} - 10 x y^{2} + 3 x y - 2 x + 10 y^{2} - 12 y + 3\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,1)$$

$$\displaystyle \phi_{3} = x y \left(10 x^{2} + 9 x y - 21 x + 10 y^{2} - 21 y + 14\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{4} = \frac{3 x \left(- 40 x^{2} y + 40 x^{2} - 11 x y^{2} + 75 x y - 64 x + 11 y^{2} - 35 y + 24\right)}{2}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{5} = \frac{3 y \left(- 11 x^{2} y + 11 x^{2} - 40 x y^{2} + 75 x y - 35 x + 40 y^{2} - 64 y + 24\right)}{2}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{6} = \frac{3 x y \left(- 11 x y + 11 x + 40 y^{2} - 53 y + 13\right)}{2}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{7} = \frac{3 x y \left(40 x^{2} - 11 x y - 53 x + 11 y + 13\right)}{2}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(2 s_{0})v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{8} = 30 x \left(2 x^{2} y - 2 x^{2} - 3 x y + 3 x + y - 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(2 s_{0})v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{9} = 30 y \left(2 x y^{2} - 3 x y + x - 2 y^{2} + 3 y - 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(2 s_{0})v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{10} = 30 x y \left(- 2 y^{2} + 3 y - 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(2 s_{0})v$$
where $$e_{3}$$ is the 3th edge;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{11} = 30 x y \left(- 2 x^{2} + 3 x - 1\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{R}(2 s_{0}^{2} - 2 s_{0} + 2 s_{1}^{2} - 2 s_{1})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{12} = 45 x y \left(- x y + x + y - 1\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$y$$, $$y^{2}$$, $$y^{3}$$, $$x$$, $$x y$$, $$x y^{2}$$, $$x y^{3}$$, $$x^{2}$$, $$x^{2} y$$, $$x^{2} y^{2}$$, $$x^{2} y^{3}$$, $$x^{3}$$, $$x^{3} y$$, $$x^{3} y^{2}$$, $$x^{3} y^{3}$$, $$\frac{7 x \left(- 5 x^{3} y + 5 x^{3} + 10 x^{2} y - 10 x^{2} - 6 x y + 6 x + y - 1\right)}{3}$$, $$\frac{7 x y \left(5 x^{3} - 10 x^{2} + 6 x - 1\right)}{3}$$, $$\frac{7 y \left(- 5 x y^{3} + 10 x y^{2} - 6 x y + x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)}{3}$$, $$\frac{7 x y \left(5 y^{3} - 10 y^{2} + 6 y - 1\right)}{3}$$
• $$\mathcal{L}=\{l_0,...,l_{19}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - 35 x^{4} y + 35 x^{4} + 100 x^{3} y^{3} - \frac{4575 x^{3} y^{2}}{26} + \frac{4055 x^{3} y}{26} - 80 x^{3} - \frac{4575 x^{2} y^{3}}{26} + \frac{4035 x^{2} y^{2}}{13} - \frac{5055 x^{2} y}{26} + 60 x^{2} - 35 x y^{4} + \frac{4055 x y^{3}}{26} - \frac{5055 x y^{2}}{26} + \frac{1163 x y}{13} - 16 x + 35 y^{4} - 80 y^{3} + 60 y^{2} - 16 y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{1} = \frac{x \left(- 910 x^{3} y + 910 x^{3} - 2600 x^{2} y^{3} + 4575 x^{2} y^{2} - 415 x^{2} y - 1560 x^{2} + 3225 x y^{3} - 5655 x y^{2} + 1650 x y + 780 x + 910 y^{4} - 2705 y^{3} + 2640 y^{2} - 741 y - 104\right)}{26}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{2} = \frac{y \left(910 x^{4} - 2600 x^{3} y^{2} + 3225 x^{3} y - 2705 x^{3} + 4575 x^{2} y^{2} - 5655 x^{2} y + 2640 x^{2} - 910 x y^{3} - 415 x y^{2} + 1650 x y - 741 x + 910 y^{3} - 1560 y^{2} + 780 y - 104\right)}{26}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,1)$$

$$\displaystyle \phi_{3} = \frac{x y \left(910 x^{3} + 2600 x^{2} y^{2} - 3225 x^{2} y - 935 x^{2} - 3225 x y^{2} + 4020 x y - 15 x + 910 y^{3} - 935 y^{2} - 15 y - 64\right)}{26}$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{4} = \frac{30 x \left(182 x^{3} y - 182 x^{3} - 221 x^{2} y^{3} + 399 x^{2} y^{2} - 568 x^{2} y + 390 x^{2} + 335 x y^{3} - 607 x y^{2} + 532 x y - 260 x - 114 y^{3} + 208 y^{2} - 146 y + 52\right)}{13}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{5} = \frac{30 y \left(- 221 x^{3} y^{2} + 335 x^{3} y - 114 x^{3} + 399 x^{2} y^{2} - 607 x^{2} y + 208 x^{2} + 182 x y^{3} - 568 x y^{2} + 532 x y - 146 x - 182 y^{3} + 390 y^{2} - 260 y + 52\right)}{13}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{6} = \frac{30 x y \left(221 x^{2} y^{2} - 335 x^{2} y + 114 x^{2} - 264 x y^{2} + 398 x y - 134 x - 182 y^{3} + 433 y^{2} - 323 y + 72\right)}{13}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{7} = \frac{30 x y \left(- 182 x^{3} + 221 x^{2} y^{2} - 264 x^{2} y + 433 x^{2} - 335 x y^{2} + 398 x y - 323 x + 114 y^{2} - 134 y + 72\right)}{13}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(2 s_{0})v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{8} = \frac{30 x \left(- 455 x^{3} y + 455 x^{3} + 221 x^{2} y^{3} - 399 x^{2} y^{2} + 1114 x^{2} y - 936 x^{2} - 279 x y^{3} + 510 x y^{2} - 816 x y + 585 x + 58 y^{3} - 111 y^{2} + 157 y - 104\right)}{13}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(2 s_{0})v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{9} = \frac{30 y \left(221 x^{3} y^{2} - 279 x^{3} y + 58 x^{3} - 399 x^{2} y^{2} + 510 x^{2} y - 111 x^{2} - 455 x y^{3} + 1114 x y^{2} - 816 x y + 157 x + 455 y^{3} - 936 y^{2} + 585 y - 104\right)}{13}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(2 s_{0})v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{10} = \frac{30 x y \left(- 221 x^{2} y^{2} + 279 x^{2} y - 58 x^{2} + 264 x y^{2} - 327 x y + 63 x + 455 y^{3} - 979 y^{2} + 633 y - 109\right)}{13}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(2 s_{0})v$$
where $$e_{3}$$ is the 3th edge;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{11} = \frac{30 x y \left(455 x^{3} - 221 x^{2} y^{2} + 264 x^{2} y - 979 x^{2} + 279 x y^{2} - 327 x y + 633 x - 58 y^{2} + 63 y - 109\right)}{13}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(3 s_{0}^{2})v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{12} = \frac{10 x \left(910 x^{3} y - 910 x^{3} - 1820 x^{2} y + 1820 x^{2} - 105 x y^{3} + 177 x y^{2} + 1020 x y - 1092 x + 105 y^{3} - 177 y^{2} - 110 y + 182\right)}{13}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(3 s_{0}^{2})v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{13} = \frac{10 y \left(- 105 x^{3} y + 105 x^{3} + 177 x^{2} y - 177 x^{2} + 910 x y^{3} - 1820 x y^{2} + 1020 x y - 110 x - 910 y^{3} + 1820 y^{2} - 1092 y + 182\right)}{13}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(3 s_{0}^{2})v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{14} = \frac{10 x y \left(105 x^{2} y - 105 x^{2} - 138 x y + 138 x - 910 y^{3} + 1820 y^{2} - 1059 y + 149\right)}{13}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(3 s_{0}^{2})v$$
where $$e_{3}$$ is the 3th edge;
and $$s_{0}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{15} = \frac{10 x y \left(- 910 x^{3} + 1820 x^{2} + 105 x y^{2} - 138 x y - 1059 x - 105 y^{2} + 138 y + 149\right)}{13}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{R}(2 s_{0}^{2} - 2 s_{0} + 2 s_{1}^{2} - 2 s_{1})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{16} = \frac{60 x y \left(- 455 x^{2} y^{2} + 735 x^{2} y - 280 x^{2} + 735 x y^{2} - 1191 x y + 456 x - 280 y^{2} + 456 y - 176\right)}{13}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{R}(6 s_{0}^{2} s_{1} - 2 s_{0}^{2} - 6 s_{0} s_{1} + 2 s_{0} + 2 s_{1}^{3} - 2 s_{1}^{2})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{17} = \frac{2100 x y \left(26 x^{2} y^{2} - 39 x^{2} y + 13 x^{2} - 42 x y^{2} + 63 x y - 21 x + 16 y^{2} - 24 y + 8\right)}{13}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{R}(2 s_{0}^{3} - 2 s_{0}^{2} + 6 s_{0} s_{1}^{2} - 6 s_{0} s_{1} - 2 s_{1}^{2} + 2 s_{1})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{18} = \frac{2100 x y \left(26 x^{2} y^{2} - 42 x^{2} y + 16 x^{2} - 39 x y^{2} + 63 x y - 24 x + 13 y^{2} - 21 y + 8\right)}{13}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{R}(6 s_{0}^{3} s_{1} - 2 s_{0}^{3} - 6 s_{0}^{2} s_{1} + 2 s_{0}^{2} + 6 s_{0} s_{1}^{3} - 6 s_{0} s_{1}^{2} - 2 s_{1}^{3} + 2 s_{1}^{2})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{19} = 2100 x y \left(- 4 x^{2} y^{2} + 6 x^{2} y - 2 x^{2} + 6 x y^{2} - 9 x y + 3 x - 2 y^{2} + 3 y - 1\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$z$$, $$y$$, $$y z$$, $$x$$, $$x z$$, $$x y$$, $$x y z$$, $$\frac{3 x \left(x y z - x y - x z + x - y z + y + z - 1\right)}{2}$$, $$\frac{3 x z \left(- x y + x + y - 1\right)}{2}$$, $$\frac{3 x y \left(- x z + x + z - 1\right)}{2}$$, $$\frac{3 x y z \left(x - 1\right)}{2}$$, $$\frac{3 y \left(x y z - x y - x z + x - y z + y + z - 1\right)}{2}$$, $$\frac{3 y z \left(- x y + x + y - 1\right)}{2}$$, $$\frac{3 x y \left(- y z + y + z - 1\right)}{2}$$, $$\frac{3 x y z \left(y - 1\right)}{2}$$, $$\frac{3 z \left(x y z - x y - x z + x - y z + y + z - 1\right)}{2}$$, $$\frac{3 y z \left(- x z + x + z - 1\right)}{2}$$, $$\frac{3 x z \left(- y z + y + z - 1\right)}{2}$$, $$\frac{3 x y z \left(z - 1\right)}{2}$$
• $$\mathcal{L}=\{l_0,...,l_{19}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

$$\displaystyle \phi_{0} = 3 x^{2} y z - 3 x^{2} y - 3 x^{2} z + 3 x^{2} + 3 x y^{2} z - 3 x y^{2} + 3 x y z^{2} - 10 x y z + 7 x y - 3 x z^{2} + 7 x z - 4 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 7 y z - 4 y + 3 z^{2} - 4 z + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v(1,0,0)$$

$$\displaystyle \phi_{1} = x \left(3 x y z - 3 x y - 3 x z + 3 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 4 y z - y + 3 z^{2} - z - 2\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(0,1,0)$$

$$\displaystyle \phi_{2} = y \left(- 3 x^{2} z + 3 x^{2} + 3 x y z - 3 x y - 3 x z^{2} + 4 x z - x - 3 y z + 3 y + 3 z^{2} - z - 2\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,1,0)$$

$$\displaystyle \phi_{3} = x y \left(- 3 x z + 3 x - 3 y z + 3 y + 3 z^{2} + 2 z - 5\right)$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(0,0,1)$$

$$\displaystyle \phi_{4} = z \left(- 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 3 x y z + 4 x y - 3 x z - x + 3 y^{2} - 3 y z - y + 3 z - 2\right)$$

This DOF is associated with vertex 4 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(1,0,1)$$

$$\displaystyle \phi_{5} = x z \left(- 3 x y + 3 x + 3 y^{2} - 3 y z + 2 y + 3 z - 5\right)$$

This DOF is associated with vertex 5 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(0,1,1)$$

$$\displaystyle \phi_{6} = y z \left(3 x^{2} - 3 x y - 3 x z + 2 x + 3 y + 3 z - 5\right)$$

This DOF is associated with vertex 6 of the reference element.
$$\displaystyle l_{7}:v\mapsto v(1,1,1)$$

$$\displaystyle \phi_{7} = x y z \left(3 x + 3 y + 3 z - 8\right)$$

This DOF is associated with vertex 7 of the reference element.
$$\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{8} = 6 x \left(- x y z + x y + x z - x + y z - y - z + 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{9} = 6 y \left(- x y z + x y + x z - x + y z - y - z + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{10} = 6 z \left(- x y z + x y + x z - x + y z - y - z + 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{11} = 6 x y \left(y z - y - z + 1\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{e_{4}}v$$
where $$e_{4}$$ is the 4th edge.

$$\displaystyle \phi_{12} = 6 x z \left(y z - y - z + 1\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{e_{5}}v$$
where $$e_{5}$$ is the 5th edge.

$$\displaystyle \phi_{13} = 6 x y \left(x z - x - z + 1\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{e_{6}}v$$
where $$e_{6}$$ is the 6th edge.

$$\displaystyle \phi_{14} = 6 y z \left(x z - x - z + 1\right)$$

This DOF is associated with edge 6 of the reference element.
$$\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{e_{7}}v$$
where $$e_{7}$$ is the 7th edge.

$$\displaystyle \phi_{15} = 6 x y z \left(1 - z\right)$$

This DOF is associated with edge 7 of the reference element.
$$\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{e_{8}}v$$
where $$e_{8}$$ is the 8th edge.

$$\displaystyle \phi_{16} = 6 x z \left(x y - x - y + 1\right)$$

This DOF is associated with edge 8 of the reference element.
$$\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{e_{9}}v$$
where $$e_{9}$$ is the 9th edge.

$$\displaystyle \phi_{17} = 6 y z \left(x y - x - y + 1\right)$$

This DOF is associated with edge 9 of the reference element.
$$\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{e_{10}}v$$
where $$e_{10}$$ is the 10th edge.

$$\displaystyle \phi_{18} = 6 x y z \left(1 - y\right)$$

This DOF is associated with edge 10 of the reference element.
$$\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{e_{11}}v$$
where $$e_{11}$$ is the 11th edge.

$$\displaystyle \phi_{19} = 6 x y z \left(1 - x\right)$$

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

## References

• Cockburn, Bernardo and Qiu, Weifeng. Commuting diagrams for the TNT elements on cubes, Mathematics of Computation 83, 603–633, 2014. [DOI: 10.1090/S0025-5718-2013-02729-9] [BibTeX]
• Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM journal on numerical analysis 55, 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]

## DefElement stats

 Element added 24 October 2021 Element last updated 10 February 2022