http://certik.github.com/ccms-10-poster/

where you can download pdf, sources, I also put there some relevant info and links.

It turned out to be a lot more work than I expected (well, as usual), but we were very thorough with John and in the process I discovered several bugs in my program, so I am glad we did it. I used to generate all the plots by hand, by manually adjusting all the parameters in the Python script (like atomic number, mesh parameters, element orders, adaptivity parameters, error tolerance and so on). Essentially I had to remember all these parameters for each of the plots (about 10 of them). Then I settled to have a Python dictionary, that holds all the parameters, and then I just pass them to a radial_schroedinger_equation_adapt(params, error_tol=1e-8) function.

Here are example of the parameters:

params_hydrogen_p_L = dict(l=0, Z=1, a=0, b=100, el_num=4, el_order=1,

eig_num=3, mesh_uniform=False, mesh_par1=20, adapt_type="p",

eqn_type="R")

params_hydrogen_p_U = dict(l=0, Z=1, a=0, b=100, el_num=4, el_order=2,

eig_num=3, mesh_uniform=True, adapt_type="p", eqn_type="R")

params_hydrogen_hp_U = dict(l=0, Z=1, a=0, b=100, el_num=4, el_order=2,

eig_num=3, mesh_uniform=True, adapt_type="hp", eqn_type="R")

params_hydrogen_h_U = dict(l=0, Z=1, a=0, b=100, el_num=4, el_order=6,

eig_num=3, mesh_uniform=True, adapt_type="romanowski",

eqn_type="rR")

params_silver_p_L = dict(l=0, Z=47, a=0, b=150, el_num=4, el_order=13,

eig_num=50, mesh_uniform=False, mesh_par1=35, adapt_type="p",

eqn_type="R")

params_silver_hp_L = dict(l=0, Z=47, a=0, b=150, el_num=4, el_order=13,

eig_num=50, mesh_uniform=False, mesh_par1=35, adapt_type="hp",

eqn_type="R")

I mean, this is kind of obvious if you think about it, but for some reason I didn't do that at all at the beginning, because I thought --- I'll just run this once and I am done with it. But I had to run it like 20x, e.g. regenerating he plots, then creating a table about meshes, then redoing the table after changing the error tolerance, and so on.

Besides that I also got permission to release my code, so I'll go over it in the coming days and generate nice patches against Hermes1D.

Also in the process of creating the poster, I played a lot with p-FEM, uniform-p-FEM, hp-FEM and h-FEM and I will keep playing with that. It's clear to me now, that our current Hermes1D is not optimal. Especially the convergence of hp-FEM and p-FEM (as it is implemented right now) greatly depends on the initial mesh.

Nevertheless, even with the above limitations, hp-FEM seems to be really good if you don't a-priori know anything about the problem/mesh. One should not make any deep conclusions in 1D (it might be a bit different in 2D and 3D, and also I only did couple test problems), but from my experience so far, hp-FEM is a really good choice, if you just want to solve the problem and get a decent convergence (way better than h-FEM, and in general about the same as uniform-p-FEM with optimized mesh).

Another conclusion is that uniform-p-FEM (also called spectral element method), if you optimize the mesh for the problem, is very fast. All you have to do is increase the polynomial order and it goes very straight on the convergence graphs, it's very hard to beat. Also, and that I would like to write in the coming days, the algorithm for optimizing the mesh is really simple: just solve it with high "p", then play with the mesh parameters (for logarithmic mesh, there are only 2 parameters --- number of elements, and a ratio of first vs. last element), so that the eigenvalues (that one is interested in) are converged (with given accuracy) and optimize it wrt DOFs. The algorithm can also "look" at the convergence graphs and make sure it's steep enough. For atomic problems, my experience shows that the logarithmic mesh is good enough (as long as you optimize it). The advantage is that you do this once, and then (for close enough potentials in the Schroedinger equation), you just increase "p", and it's very robust and fast (no need for reference mesh, or trial refinements and so on).

When I get back to Reno, we'll do more research on hp-FEM with Pavel and I think this is not the last word to say. We need to review how we choose the candidates for eigenvectors, especially "p" vs "hp" and make it more robust. We'll see.