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 \(\text{curl} \, \mathbf{u} = \nabla\cdot R \mathbf{u}\).
These two de Rham complex definitions in 2D account for the double element diagrams in the orange boxes in the Periodic table of the finite elements. That is also why the boxes are orange, and not red nor yellow as in 3D.