I'm trying to understand the advantages of the Helmholtz-Hodge Decomposition (HHD) in detecting singularities (sources, sinks, centers of rotation) of a vector field in a discrete setting. Since I am not a mathematician and my quantitative background is quite limited, I'd appreciate more of an intuitive/conceptual answer.
Polthier and Preuß (2003; Identifying vector field singularities using a discrete Hodge decomposition) have used a discrete Hodge decomposition to identify singularities in a vector field. As far as I understand they calculate the potentials of the HHD by solving the Poisson equations
$\Delta D = div(\vec{v})$,
$\Delta R = curl(\vec{v})$
where $D$ and $R$ are the potentials associated with the curl- and divergence-free components of the vector field ${v}$ (Bhatia et al., 2013; The Helmholtz-Hodge Decomposition—A Survey). Next, to find sources, sinks and centers of rotation, they identify local extrema of those potentials.
In order to compute those potentials, one also needs to compute the divergence and curl (e.g. using eqs. 9 here). Why can we not directly find the extrema of $div(v)$ and $curl(v)$ instead of identifying them in the HHD potentials? I would appreciate any intuitive explanation of how the HHD is advantageous in this case.
Furthermore, if I calculate the gradient of a scalar field $U$ defined on the vertices of a triangulated mesh and compute the HHD of the resulting gradient field, the scalar potential $D = U$ while $R = 0$ everywhere, or am I misunderstanding this? Thus, if one starts with a scalar field the corresponding gradient field's singularities are the extrema of that scalar field.