Multigrid methods are linear solvers. Generally they are only not used in the context of timestepping. PDEs will be a separate post.

I'm so looking forward to this post!. hp-FEM methods, in particular, caught my attention, they look so promising. Do they work in practice?

Some time ago I was trying to make a 'hpq-FEM' using rational functions as a basis, like a Padé approximation. Do you know if anyone has tried something similar?