MueLu: PCoarsen factory extension for serendipity element

[nasseralkmim]

Question

Hi, I would like to try the Pcoarsen factory to create a 2-level hierarchy from a quadratic element to a linear element.

I noticed that Serendipity quad elements were introduced here https://github.com/trilinos/Trilinos/commit/c3b1a94753b3611e3f931a1a1b6e9f026b30ecb8.

But the interface in MueLu apparently is not updated to parse those https://github.com/trilinos/Trilinos/blob/fb58dc8b5611d9606664cebd251de0b8949892ed/packages/muelu/src/Transfers/PCoarsen/MueLu_IntrepidPCoarsenFactory_def.hpp#L180. Not sure if the rest of the PCoarsen factory is capable of supporting those elements.

Some questions:

1. How to extend the Pcoarsen factory to support serendipity elements? (at least the BasisFactory seems straight forward).
2. Is there an example of setting up a prolongation operator for transferring elements with two dofs per node? (the QUAD element has one dof per node https://github.com/trilinos/Trilinos/blob/master/packages/intrepid2/src/Discretization/Basis/Intrepid2_HGRAD_QUAD_C2_FEM.hpp , maybe with DerivedClasse?
3. Is it possible to set up a 2-level hierarchy with PCoarsen prolongation, and as a coarse-level solver another hierarchy with MueLu? I know we can use Teko as a smoother, and we can use MueLu within Teko, so this may be an alternative.

Best regards, Nasser