Complex families

You can find some information about how these familes are defined here

De Rham complex in 3D

Name(s)	\(H^1\)	\(\xrightarrow{\nabla}\)	\(H(\textbf{curl})\)	\(\xrightarrow{\nabla\times}\)	\(H(\text{div})\)	\(\xrightarrow{\nabla\cdot}\)	\(L^2\)
\(S_{1,k}^\unicode{0x25FA}\), \(\mathcal{P}_{k}\Lambda^{r}(\Delta_3)\)	Lagrange		Nédélec (second kind)		Brezzi–Douglas–Marini		Lagrange
\(S_{2,k}^\unicode{0x25FA}\), \(\mathcal{P}^-_{k}\Lambda^{r}(\Delta_3)\)	Lagrange		Nédélec (first kind)		Raviart–Thomas		Lagrange
\(S_{4,k}^\square\), \(\mathcal{Q}^-_{k}\Lambda^{r}(\square_3)\)	Lagrange		Nédélec (first kind)		Q H(div)		Lagrange
\(S_{1,k}^\square\), \(\mathcal{S}_{k}\Lambda^{r}(\square_3)\)	serendipity		serendipity H(curl)		serendipity H(div)		dPc
\(S_{2,k}^\square\), \(\mathcal{S}^-_{k}\Lambda^{r}(\square_3)\)	serendipity		trimmed serendipity H(curl)		trimmed serendipity H(div)		dPc
\(S_{3,k}^\square\)	Tiniest tensor		Tiniest tensor H(curl)		Tiniest tensor H(div)		Lagrange

Name(s)	\(H^1\)	\(\xrightarrow{\textbf{curl}}\)	\(H(\text{div})\)	\(\xrightarrow{\nabla\cdot}\)	\(L_2\)
\(S_{1,k}^\unicode{0x25FA}\), \(\mathcal{P}_{k}\Lambda^{r}(\Delta_2)\)	Lagrange		Brezzi–Douglas–Marini		Lagrange
\(S_{2,k}^\unicode{0x25FA}\), \(\mathcal{P}^-_{k}\Lambda^{r}(\Delta_2)\)	Lagrange		Raviart–Thomas		Lagrange
\(S_{4,k}^\square\), \(\mathcal{Q}^-_{k}\Lambda^{r}(\square_2)\)	Lagrange		Q H(div)		Lagrange
\(S_{1,k}^\square\), \(\mathcal{S}_{k}\Lambda^{r}(\square_2)\)	serendipity		serendipity H(div)		dPc
\(S_{2,k}^\square\), \(\mathcal{S}^-_{k}\Lambda^{r}(\square_2)\)	serendipity		trimmed serendipity H(div)		dPc
\(S_{3,k}^\square\)	Tiniest tensor		Tiniest tensor H(div)		Lagrange
\(\mathrm{C}\mathrm{P}^{3-r}\mathrm{\Lambda}^r(\mathcal{R})\)	Hsieh–Clough–Tocher		Alfeld–Sorokina		P1 macro