This week I have been wrapping up my work at LLNL and trying to generate some comparisons between different approaches to adapt to multiple eigenvectors at once.
It turns out that most of the issues are in the way we create the new mesh for the next adaptivity iteration, in particular, how do we choose which candidates to refine. I have tried to converge to the lowest eigenvector (as well as to any other eigenvector too), to the sum of the eigenvectors, to each eigenvector individually and taking the union of the meshes and so on. I have also implemented the uniform p-FEM as well as tried p-FEM and hp-FEM using the approaches I mentioned above.
It seems to be crucial to have a good initial mesh, at least for p-FEM. If I use a good mesh and p-adaptivity, I am able to get the best results so far. In principle hp adaptivity should be at least as good, but our current approach doesn't show it yet. Hopefully we'll manage to make it work.
Besides that I have also implemented H1 norms for the Function class (based on Fekete points) by calculating coefficients with regards to a FE basis and some other little things.