I did something similar for the LLG equation. I have a time and space adaptive code, which is doing the refinement in space not with the AdaptiveNonlinearSolver but by refining the mesh based on some residual error estimator. The “coarsening” itself is no real coarsening but a remeshing in the sense that when ever we want to coarsen the mesh we refine a coarse mesh until it fits the current solution. Code is available on github and may be adapted to other equations than the LLG.
janbohn/AdaptiveLLG: space and time adaptive algorithm for the LLG equation
you may also consider
Hierarhical mesh - mesh - FEniCS Project
and the linked Github from Alexander Zimmerman