an encyclopedia of finite element definitions
You can find some information about how these familes are defined here
| Name(s) | \(H^1\) | \(\xrightarrow{\nabla}\) | \(\textbf{H}(\text{curl})\) | \(\xrightarrow{\nabla\times}\) | \(\textbf{H}(\text{div})\) | \(\xrightarrow{\nabla\cdot}\) | \(L^2\) |
| \(S_{1,k}^\unicode{0x25FA}\), \(\mathcal{P}_{k}\Lambda^{r}(\Delta_3)\) | Lagrange | Nédélec (second kind) | Brezzi–Douglas–Marini | discontinuous Lagrange | |||
| \(S_{2,k}^\unicode{0x25FA}\), \(\mathcal{P}^-_{k}\Lambda^{r}(\Delta_3)\) | Lagrange | Nédélec (first kind) | Raviart–Thomas | discontinuous Lagrange | |||
| \(S_{4,k}^\square\), \(\mathcal{Q}^-_{k}\Lambda^{r}(\square_3)\) | Lagrange | Nédélec (first kind) | Raviart–Thomas | discontinuous 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) | discontinuous Lagrange |
In 2D, \(\textbf{H}(\text{div})\) and \(\textbf{H}(\text{curl})\) are isomorphic via a 90 degree rotation \(R\). This means that we can define the de Rham complex in two ways.
One variant uses \(\textbf{H}(\text{div})\) and the vector-valued \(\textbf{curl}\) operator, \(\textbf{curl} \, u = (\partial_y u, -\partial_x u)\). Note that \(\textbf{curl} \, u = R \, \nabla u\).
The second variant uses \(\textbf{H}(\text{curl})\) and the scalar-valued \(\text{curl}\) operator, \(\text{curl} \, \mathbf{u} = \partial_x u_y - \partial_y u_x\). Note that \(\text{curl} \, \mathbf{u} = \nabla\cdot R \mathbf{u}\).