[Dune] Non-unique coordinate mappings of Codim(1) entities?

[Christian Waluga]

Hi,

Am 07.06.2010 um 19:20 schrieb Christian Engwer:
>> I am trying to implement  a hybrid DG method using DUNE and I have a
>> problem with the mapping feature of the intersections. In addition
>> to my element DOFs, I have an additional function living on the
>> trace of the grid, i.e. I have some DOFs distributed on the faces of
>> the mesh. The intersections of the elements seem to be associated
>> with the codim 1 entities of the mesh, but the orientation of their
>> geometries seems to be not unique in my experiments (using
>> ALUSimplexGrid<2,2>).
> 
> no, the intersections do not (necessarily) correspond to the codim1
> entities. Intersections are of codim1, but they are not an entity of
> the grid. 

Yes, that's true. I think I see where the problem is, now that you gave an example with a nonmatching mesh. In the nonmatching case, the intersections can map to the inside and outside geometry, but the codim 1 entities e1 and e2 in my example might indeed differ.

My problem now is the following: I need to define finite element functions on the trace (they couple with the element-wise discontinuous functions in the hybrid DG method). But intersections in Dune seemed to me like temporary element-specific objects, so I did not consider to create a map that does an indexing of intersections to compute the local to global mapping of the DOFs. Therefore I defined my DOFs on the codim 1 entities, but this will definitely fail, if the mesh is nonmatching and I'm not very comfortable with my solution.

So, intersections are not entities of the grid, but codim 1 entities are also not the entities I want to define my trace finite element space on. Intersections can also not be easily indexed, as far as I know... My idea to solve this is to create a map that identifies the 'unique' intersection from the pair indices of is.inside() and is.outside(). So e.g. intersection 1 is between elements 2 and 3, intersection 2 is between elements 1 and 2, etc... What remains is the problem with the orientations in the coordinate mappings of the intersection-geometries in the 'inside' and 'outside' element. Or is there a more elegant way to construct a trace finite element space (dofs living on the intersections) like the one I need? I have a feeling that the solution might be easier than I think.

Thanks for any response!

Best, Chris