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

[Claus-Justus Heine]

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Dear Christian,

I guess that somehow sticking to one fixed mesh-concept -- as naturally
enforced in ALBERTA -- seems to  make things look more complicated as
they are, from an implementation oriented point of view.

On 19.03.2014 00:59, Christian Engwer wrote:> Dear Claus-Justus,
[snip]

> Following the meta-grid philosophy, the trace-grid would never do the
> refinement on it's own, but mark the bulk-entity for refinement. After
> the adaptation we obtain the refinement of the trace entity.
>
> I'd imagine the following:
[snip]

Well, thats ("of course") exactly like things are handled in Alberta.

For theoretical reason it is probably a nice feature of simplicial
meshes generated by newest-vertex recursive bisection refinement that
the trace-meshes form admissible mesh instances in their own right.
However, I agree that ...

> ... it is simple for conforming functionspaces, as you don't have to
> deal with intersections and just ommit them, but otherwise the missing
> intersections will require a substential effort.

... those non-conforming intersections would be a nightmare concerning
implementation (oh well, not quite a nightmare. But the implementation
would be quite technical).

Overall, for conforming meshes (i.e. no hanging nodes) and conforming
(i.e.: "even conforming") Lagrange spaces (including tensor-product
spaces) the implementation would be more or less straight forward
(omitting the additional task to match the trace-mesh to one of the
"base" grid implementations, which probably is not even desirable).

Concerning applications:
The "need" for exact traces of finite element space traces for "real
life" applications is of course some other issue; typically -- in my
imagination -- trace-meshes and trace-FEM-spaces would occur in a
context where one tries to enforce a strong coupling between solutions
of PDEs defined at the boundary to bulk quantities. Whether such a
strong coupling is needed/not needed/desirable/unfortunate/whatever is
another story.

So, many thanks for all your comments! Until soon.

Claus

- -- 
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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