an encyclopedia of finite element definitions

# De Rham element families

Written by Matthew W. Scroggs
The following relationship is the $$L^2$$ de Rham complex: $$H^k \xrightarrow{\nabla} H^{k-1}(\textbf{curl}) \xrightarrow{\nabla\times} H^{k-1}(\text{div}) \xrightarrow{\nabla\cdot} H^{k-1}$$
A set of four finite elements $$\mathcal{V}_0$$ to $$\mathcal{V}_3$$ forms a discrete de Rham complex if the following commitative diagram holds, where $$I_0$$ to $$I_3$$ are interpolations into $$\mathcal{V}_0$$ to $$\mathcal{V}_3$$. (The commutative diagram holds if following different arrow combinations to the same destination will give the same result.) $$\begin{array}{ccccccc} H^k &\xrightarrow{\nabla} &H^{k-1}(\textbf{curl}) &\xrightarrow{\nabla\times} &H^{k-1}(\text{div}) &\xrightarrow{\nabla\cdot} &H^{k-1}\\ \hphantom{\small I_0}\big\downarrow {\small I_0}&& \hphantom{\small I_1}\big\downarrow {\small I_1}&& \hphantom{\small I_2}\big\downarrow {\small I_2}&& \hphantom{\small I_3}\big\downarrow {\small I_3}\\ \mathcal{V}_0 &\xrightarrow{\nabla} &\mathcal{V}_1 &\xrightarrow{\nabla\times} &\mathcal{V}_2 &\xrightarrow{\nabla\cdot} &\mathcal{V}_3 \end{array}$$