# Why does the naive barycentric hodgestar fail?

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. (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?