Korinna,

> 1) Maybe I found a wrong assertion in the FETools.
>     I have trouble with FETools while working in the debug mode. I cannot
> use the provided FETools::interpolation for the higher order reconstruction,

It might be that we haven't been testing this function enough, so a
bug like this is not unlikely.

>        The violated condition was:
>            owned_elements.size()==size()
>        Additional information:
>           The locally owned elements have not been properly initialized!
> This happens for example if this object has been initialized with exactly
> one overlapping IndexSet.

Can you try initializing your ghosted vector with
vec.reinit(locally_owned_index_set, locally_relevant_index_set, mpi_com);
instead of only specifying the ghosted set? Maybe that fixes the exception.

> 2) I have also a much worse problem right now, for which I have not found a
> satisfying solution yet.
>    It seems to me that I cannot use the MeshSmooting methods for the
> parallel::distributed::Triangulation. In particular the flag patch_level_1
> causes trouble.

Yes, mesh smoothing in parallel is a problem at the moment. It is
basically not working at all. Sorry. I was hoping to have an
implementation for that soon (we did some progress towards it last
month). See 
https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_dealii_dealii_pull_1711&d=DwIBaQ&c=Ngd-ta5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=4k7iKXbjGC8LfYxVJJXiaYVu6FRWmEjX38S7JmlS9Vw&m=fx5zPVPj5YHLwSgXHBKNn1I9OvTImEnlVgDaDmSH-Z0&s=xMjuyhQn_tZbaM_K32G208xiQJ7EjMmKAmSRfDoy_4Y&e=
 
https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_dealii_dealii_issues_1655&d=DwIBaQ&c=Ngd-ta5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=4k7iKXbjGC8LfYxVJJXiaYVu6FRWmEjX38S7JmlS9Vw&m=fx5zPVPj5YHLwSgXHBKNn1I9OvTImEnlVgDaDmSH-Z0&s=HmN6gJQSWE1WxMwrHI5csg3_TaC8JnCXyOa6GAnnxGQ&e=
  and issues linked in
there.
One workaround for now might be to switch to a shared::Triangulation.

Best,
Timo

-- 
Timo Heister
http://www.math.clemson.edu/~heister/

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