enable coarse multigrid mappings of low mapping degree

[nfehn]

addresses #615.

In upcoming commits, I want to introduce the capability to use a different mapping degree for the coarse multigrid mappings.

[nfehn]

The problem for the test case from #614 with boundary layer manifold, which is critical in terms of the mapping degrees, can be overcome by setting this->param.mapping_degree_coarse_grids = 1 using the capabilities introduced in the present PR.

@kronbichler @peterrum Regarding the "variational crime" mentioned here #615 (comment): As we have seen, mapping degree of 2 for all levels does not work. Interestingly, I have just realized that mapping degree 3 on all levels seems to be fine for this setup. This gives a point of reference regrading the degradation in iterations counts when using linear mappings for the coarser multigrid levels. Mapping degree 3 for all levels requires 24.8 iterations, while mapping degree 3 for the fine level and a linear mapping for all coarser levels requires 33.3 iterations, always using pure h-MG! So the increase in iterations counts seems to be substantial here. For cph-MG, iterations counts are 6.9 for both mapping setups, so no difference is visible!

[kronbichler]

These numbers are an interesting observation. I think there are two different mechanisms at work regarding the multigrid convergence here:

• Changing the mapping degree changes the approximated operators, and thus, they have (somewhat) different eigenvalue spectra, such that the decomposition in different scales with smoothers damping the "high-frequency" content has a different meaning. In the extreme case, having a bad mapping (like the degree 2 case you mention) means that the eigenvalue spectrum of the matrix A gets bad.
• The mapping can also make the standard embedding of the multigrid transfer, that is defined in the reference element, a poor choice. This means that even if the coarse-scale solution might be good "correction", the embedding of it into the fine level might distort it. I would intuitively say that this should be a fine-scale/high-frequency perturbation, so the fine-level smoother should be good at removing it again, but eigenmodes resulting from polynomial clipping might do badly.

Since the difference you observe is substantial, I think it would instructive to try to separate the two issues. I think we should be able to use the non-matching transfer operator for testing, which would make the embedding the natural one in the expanded base. That would then level my first item.