an encyclopedia of finite element definitions

De Rham element families

Written by Matthew W. Scroggs
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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.) $$ \begin{array}{ccccccc} H^k &\xrightarrow{\nabla} &H^{k-1}(\textbf{curl}) &\xrightarrow{\nabla\times} &H^{k-1}(\text{div}) &\xrightarrow{\nabla\cdot} &H^{k-1}\\ \hphantom{\small I_0}\big\downarrow {\small I_0}&& \hphantom{\small I_1}\big\downarrow {\small I_1}&& \hphantom{\small I_2}\big\downarrow {\small I_2}&& \hphantom{\small I_3}\big\downarrow {\small I_3}\\ \mathcal{V}_0 &\xrightarrow{\nabla} &\mathcal{V}_1 &\xrightarrow{\nabla\times} &\mathcal{V}_2 &\xrightarrow{\nabla\cdot} &\mathcal{V}_3 \end{array} $$

You can view families of elements that form discrete de Rham complexes on the families page. On DefElement, two naming conventions for elements in a de Rham complex are used. The first of these is the exterior calculus convention: this is the notation used in the Periodic table of the finite elements[1]. The second is the Cockburn–Fu convention: this gives the names used for element families in Cockburn and Fu's 2017 paper[2].