Implement VectorTools::interpolate_to_finer/coarser_mesh().

[bangerth]

I mentioned this on the elements forum: I'm in need of a function that can interpolate between coarse and fine DoFHandler objects not only on a single process, but indeed in parallel where the partitioning of the two meshes may not geometrically match. This is difficult to implement, but as @kronbichler and @peterrum mentioned, they have already provided this kind of functionality in the form of the MGTwoLevelTransfer class :-)

This patch makes use of this to implement the functions needed. The majority of the code in fact is due to the limitation that MGTwoLevelTransfer can only deal with parallel::distributed::Vector objects (see #16684). I decided that I was more interested in getting something to work for now than to make it super efficient, and so I just go with this limitation and copy data back and forth between the user-side vector types and the internally required parallel::distributed::Vector objects.

[kronbichler]

I think this looks good, apart from some small comments.

I have an additional question: I think we can also address some of

dealii/include/deal.II/fe/fe_tools.h

Lines 621 to 654 in 54ef806

	* Compute the interpolation of a the @p dof1-function @p u1 to a @p
	* dof2-function @p u2. @p dof1 and @p dof2 need to be DoFHandlers based on
	* the same triangulation.
	*
	* If the elements @p fe1 and @p fe2 are either both continuous or both
	* discontinuous then this interpolation is the usual point interpolation.
	* The same is true if @p fe1 is a continuous and @p fe2 is a discontinuous
	* finite element. For the case that @p fe1 is a discontinuous and @p fe2 is
	* a continuous finite element there is no point interpolation defined at
	* the discontinuities. Therefore the mean value is taken at the DoF values
	* on the discontinuities.
	*
	* Note that for continuous elements on grids with hanging nodes (i.e.
	* locally refined grids) this function does not give the expected output.
	* Indeed, the resulting output vector does not necessarily respect
	* continuity requirements at hanging nodes: if, for example, you are
	* interpolating a Q2 field to a Q1 field, then at hanging nodes the output
	* field will have the function value of the input field, which however is
	* not usually the mean value of the two adjacent nodes. It is thus not part
	* of the Q1 function space on the whole triangulation, although it is of
	* course Q1 on each cell.
	*
	* For this case (continuous elements on grids with hanging nodes), please
	* use the @p interpolate() function with an additional AffineConstraints
	* object as argument, see below, or make the field conforming yourself
	* by calling the @p distribute function of your hanging node constraints
	* object.
	*/
	template <int dim, int spacedim, class InVector, class OutVector>
	void
	interpolate(const DoFHandler<dim, spacedim> &dof1,
	const InVector &u1,
	const DoFHandler<dim, spacedim> &dof2,
	OutVector &u2);

and similar functions in FETools with this infrastructure. This does not need to happen now, but I think it is good to have this in mind, to simplify life for users in this regard in the near future.

[kronbichler]

You don't need this, do you?

Suggested change

my_u_coarse.update_ghost_values();

[bangerth]

You're right, I don't!

[kronbichler]

Suggested change

my_u_fine.update_ghost_values();

[peterrum]

Let me point out that https://www.dealii.org/developer/doxygen/deal.II/classDataOutResample.html and https://www.dealii.org/developer/doxygen/deal.II/classMGTwoLevelTransferNonNested.html is for this purpose.

[bangerth]

@kronbichler I removed the two lines. For the functions in FETools: Good grief, they look exactly like the functions in VectorTools. I had not remembered that we have this stuff twice. I'll leave a couple links in a separate PR (#16743).