Raviart–Thomas elements

Remarks

Lowest order on a triangle: $RT_0(T) = \set{x \mapsto a + b x : a \in \R^2, \ b \in \R}$, of dimension $3$. On an edge $e$ with unit normal $n$, the points satisfy $x \cdot n = c$ for a constant $c$. So $(a + b x) \cdot n = a \cdot n + b c$ is constant on $e$, and it is fixed by the flux $\int_e v \cdot n$. The three fluxes determine $v$, which gives the three degrees of freedom.

The dimension of $RT_k(T)$ on a triangle is $(k + 1)(k + 3)$. Matching the normal moments makes the normal component continuous across faces. So the global space lies in $H(\operatorname{div})$, the $L^2$ fields with divergence in $L^2$. The divergence maps $RT_k(T)$ onto $P_k(T)$. Paired with discontinuous piecewise polynomials of degree $k$, these spaces give a stable mixed finite element method for the Poisson equation. They were introduced by Pierre-Arnaud Raviart and Jean-Marie Thomas.