The classical de-Rahm complex is an important tool in the analysis and discretization of, e.g., Maxwell's equations, and is the prototypical Hilbert complex. There, each space in the exact sequence is connected via the exterior derivative.
When we now turn to hyperbolic metrics, one needs to twist the $L^2$ inner product to preserve the Hilbertian structure. This gives rise to an adjoint exterior derivative with its own Hilbert complex. As it turns out, these complexes are linked via a geometric operation. In this talk, we will introduce those exterior derivatives, explain how they are linked, and how this aids in the analysis and discretization of hyperbolic problems.
We will conclude with first numerical results, which indicate unconditional stability of the resulting discretization.
Hopefully, this talk serves as a starting point for later discussions of the open points.