Figure 6.

The various discrete Hodge stars depend on different aspects of mesh geometry as shown in this 2D examples. The diagonal Hodge star (left) computes ratios of sizes of primal-dual element pairs. The Whitney Hodge star (middle) has entries of Whitney functions integrated against each other. The support of a particular ${\mathcal{\text{W}}}_{{\sigma }_i^1}$ function is shown in grey; the integral of its projection to the bold edge has value 1. The Dual Hodge star (right) that we propose has entries of dual Whitney functions integrated against each other. The support of a particular ${\overline{\mathcal{\text{W}}}}_{\star {\sigma }_i^1}$ is shown in blue; the integral of its projection to the bold dual edge has value 1.