[Dune] Trace-Meshes like ALBERTA "Sub-Meshes" (which are trace meshes, i.e. codim 1)

[Claus-Justus Heine]

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On 18.03.2014 22:28, Andreas Dedner wrote:
> Hi.

Do you ever sleep? ;) Marveling.

> Actually I would love to have that as well. But I do not think there
>  yet is a "metagrid" is this type available. But perhaps
> somebdune at dune-project.orgody is working on this?
>
> I do not think the conformity etc. is the problem - the trace mesh
> would be a dune grid in any case. The metagrid would simply only be

Yes.

> conforming if the underlying grid has some special property. But in
> principal there is no issue. I guess implementing the intersections
> on the tracegrid would be the main work which would have to done.

- From the implementation, yes, because relative to the bulk-mesh these
would be co-dim 2 intersection which cannot be inherited from the bulk-mesh.

There would be more problems to settle.

Of course, your are right: for one fixed volume grid it is "easy"
("only" a matter of work-hours) to define a codim-1 section through the
bulk mesh as a Dune mesh in its own right. And I very happily agree that
this is/would be an outcome of one of the strength of the very generic
geometry layout of Dune (a codim-1 entity is still a geometry, with an
attached reference simplex and so on).

However, problems arise when it comes to adaptivity in the context of
conforming meshes. Ideally, this should work in both directions, and
there the interesting part of the story starts. Does an arbitrary
adaptation of the trace-mesh leads to an admissible refinement of the
bulk mesh? What about matching degrees of freedom and finite element
spaces (such that both finite element spaces remain "standard" in a
to-be-settled context)? Ideally, one would hope to have a match
between the trace-mesh and any of the existing dune-grid implementations
such that the potentially distinct refinement algorithms of Nd and
(N+1)d are compatible. For which of the existing meshes does this hold?

However, restricting my question to a more realistic context, then the
answer is: not implemented yet. But for simplicial meshes based on
recursive bisection in 2d and 3d it would be a mere matter of investing
the time necessary to implement that feature. Where I still think that
- -- talking about conforming meshes -- the crucial part is that
adaptation should work both ways (initiated either from the bulk or the
trace mesh)

Best,

Claus

>
> Andreas
>
> On 18.03.2014 22:22, Claus-Justus Heine wrote:
>> Hi there,
>>
>> recently I have been asked whether there is some Dune facility, add
>> on, project, grid stuff which has a functionality comparable to
>> Alberta's "Sub-Meshes" (would better have been labeled:
>> trace-meshes).
>>
>> Ok, for Alberta (1d, 2d, 3d simplex meshes with recursive
>> bisection) this is "easy": there is some theoretical overhead which
>> ensures that any trace-mesh (e.g. the 2d boundary mesh of a 3d
>> triangulation) is auto-"magically" an admissible 2d mesh, trivially
>> this holds also for the 1d trace-meshes derived from 2d surface
>> meshes. Also, the affine closures generated by either refining the
>> trace mesh are somehow compatible to the affine closures generated
>> by a mere bisection of the trace mesh. Additionally, one can then
>> define the respective traces of finite element spaces and it shows
>> that all this works out quite nicely.
>>
>> Is there some Dune equivalent/something for this?
>>
>> Please do not point me to the dune-subgrid module, as this would
>> be completely off-topic. Despite of the somewhat matching names
>> dune-subgrid deals with complete other stuff. ALBERTA's sub-meshes
>> define traces, whereas the subgrid module of Dune treats a
>> selection of bulk-elements as a new mesh in its own right. (no
>> offend meant, and also I suspect that consistent trace-meshes +
>> trace spaces in the context of standard, i.e. continuous finite
>> elements, are probably a specialty of certain simplicial meshes,
>> taking the refinement algorithm into account as well)
>>
>> Asking just out of curiosity.
>>
>> Kind regards,
>>
>> Claus
>>
>>
>>
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>
>
>
>
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- -- 
Dr. Claus-Justus Heine
Institut für Angewandte Analysis und Simulation
* Numerische Mathematik für Höchstleistungsrechner
Universität Stuttgart

Fon: +49 (0) 711 - 685 65558
Fax: +49 (0) 711 - 685 65507
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