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The discrete exterior calculus is defined first using circumcentric dual cells, because the primal and dual edges are orthogonal and thus the dual cells are convex. This leads to a diagonal hodge star for 1-forms that is defined as $\star_1 = \frac{|\star\sigma^1|}{|\sigma^1|}$ for a primal edge $\sigma^1$ and a dual edge $\star\sigma^1$, what has many computational advantages.

For barycentric cells, the operators are more complicated, constructions for $\star_1$ are found for example in this paper by Mohamed, Hirani and Samtaney. Their hodge star has no sparse inverse and while they suggest how a sparse inverse could be constructed, it would not satisfy the identity $\star_1\star_1^{-1} = -1$.

The reason for the more complicated constructions is, that the dual edges are no longer orthogonal and are kinked at their intersection with the primal edges.

a barycentric and a circumcentric dual mesh

(Image by Jiansong Deng)

I am now wondering, why you cannot construct a diagonal hodge star by intergrating over the kinked lines. The relevant identities are

$$ \langle\omega^1, \sigma^1\rangle = \langle\star\omega^1, \star\sigma^1\rangle \\ \langle\text{d}\omega^1, \sigma^2\rangle = \langle\omega^1, \partial\sigma^2\rangle \\ \langle\text{d}(\star\omega^1), \star\sigma^0\rangle = \langle(\star\omega^1), \partial(\star\sigma^0)\rangle $$

For a circumcentric dual 2D mesh $\star\omega^1$ represents are straight line that connects the circumcenter of two triangles and intersects the center of the primal edge.

In a barycentric mesh it is the combination of two line segments. The first identity should hold by definition, when we construct the hodge star by integrating over both segments. And I think Stokes' theorem should hold when integrating over the whole boundary of a barycentric 2-cell.

Which part am I missing, why you cannot construct a diagonal definition that allows for non-convex boundaries composed of split edges with a kink in the middle?

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