Usefulness of higher dimensional finite elements: State spaces

[luiswirth]

It would be important to highlight the usefulness of finite elements in higher dimensions.

One promising route might be the discretization of state spaces. In contrast to real space (which is almost never more than 3D), it's very common to have state spaces that are high dimensional. PDEs posed on these state spaces can then be discretized using arbitrary dimension FEEC.

Why would it be useful to use FE instead of just FD (finite difference)? The selling point of FE remains valid: You can use more complicated meshes, which are more refined at important regions of space and more coarse in irrelevant parts of the domain.
If one knows that a certain region of state space will be visited very often or requires high resolution, then we can just use a more refined mesh for this region. This is a more efficient approach than FD would offer.

A potential candidate for such a state space PDE, is the Schrödinger equation, which is posed over state space. So when using 2 particles in 3d space, we already have a PDE in 6D. This could be a nice demonstration of this ability of ADFEEC.