luiswirth
commented
2 days ago As far as I understand it, this involves solving the PDEs on a Riemannian manifold $(M, g)$, which is a smooth manifold $M$ with an inner product $g_p$ on the tangent space $T_p M$ at each point $p \in M$. This inner product is called the Riemannian metric, but it's not a metric in the sense of a metric space. It's a special case of a metric tensor. It gives rise to geometric notions such as length, area, volume and so on.
For this I need to mesh the manifold, but I do this intrinsically, meaning this is a coordinate-free formulation, meaning the vertices have no fixed coordinates, but instead I have a Riemannian metric that defines the geometric properties. This then takes effect in the Hodge star operator that is used inside the Hodge-Laplace operator.
Instead of embedding the simplicial complex, we can implement everything with the intrinsic manifold view. This means that the vertices no longer have coordinates, but all the geometry information stems from the edge lenghts. We can use the regge metric for this.