Thanks for the detailed explanation. It is quite helpful. However, I am
wondering if you could elaborate on implementation side a little bit more.
(A comment first: I think in the second picture, you wanted to use only
> two red DoFs, not three, on the common edge, right?)
Yes, that is right and I believe that it is how dealii originally implement
FE_FaceQ after enforcing the constraints.
If you *had* to implement something like this, then the way to do it
> would probably be to implement the space that corresponds to your top
> example and use it on *all* edges. Then you would create hanging node
> constraints as usual, and after that you would loop over the mesh and
> add constraints yourself for all edges that are not further refined;
> these constraints would bring the 2+2 DoFs back down to the two you want
> on these edges.
>
> This approach is not convenient, because it's not already implemented in
> deal.II, but it probably wouldn't be terribly hard to implement. You'd
> have to re-implement something like FEFaceQ, though, because you now
> want the space to not be defined on a face, but on each of its children
> (whether the edge is refined or not doesn't actually matter here).
>
It is not clear to me how modifying FEFaceQ can even achieve the first step
(highlighted in yellow) you describe. It seems that FEFace_Q is universally
defined on *the unit face* and cannot see the structure of triangulation.
Hence, the finite element space is always defined on the each face of an
element when we are doing "distribute_dofs". Are you referring that we can
use DSSY element (given in the paper) to construct the discontinuous
function defined on the reference domain of a face to achieve the first
step?
An possible way I imagine is to re-defined FEFace_Q in the following way to
complete the first step:
(assume degree = 1, so fcn 1 and fcn 2 are Lagrange polynomial with degree
1)
fcn1 fcn2
*------------**------------* <- basis function with supporting points *
o - - - - - - - - - - - - - o <- geometric line segment in reference
domain with vertices o
However, it seems not possible since the construction of a finite element
space relies on inheritance of "FE_PolyFace" and such discontinuous
function is not polynomial.
Any further comment or suggestion is greatly appreciated.
Best Regards,
Jau-Uei Chen
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