an encyclopedia of finite element definitions

# 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