[dune-functions] Basis with nodes on different gridviews

[Simon Praetorius]

Hi,

while reading the dune-functions arxiv paper, I was thinking: does it
make sense to allow basis nodes to be defined on different grids (or
gridviews)? Let's say we have a domain G and two different
triangulations T1 and T2 of G. Speaking of a global basis, e.g.
composite<lagrange<1,T1>, lagrange<1,T2>>, each lagrange node has a
number of basis functions that might be different for T1 and T2. But
this difference does not really matter for an indexing of the basis
functions of the composite basis, right? Not all index merging
strategies make sense, but it does not make sense for other composite
bases, as well. So, consider a (flat/blocked)lexicographic merging. This
should be fine, right?

But the conceptual difficulty is: what is a the corresponding local
basis? What does "local" mean in this context at all?

There are some finite-element libraries supporting something like a
multi-mesh, i.e. a collection of different refinements of a coarse-grid.
These are not two completely different triangulations, but
triangulations that allow some kind of union, in the sense that the
refinement trees can be merged. This would lead virtually to a new grid
with a refinement hierarchy containing the hierarchies of both
triangulations T1 and T2. With this concept one may introduce a
localization of the global basis to this virtually united grid, where
each basis node may be bound to an entity in the hierarchy of the
virtual grid above the current local entity contained in the
corresponding gridview of the node.

Does this makes sense? In the implementation of this extensions, there
are some difficulties. But a very rudimentary is done, that allows
already the interpolation into this basis. The main problem is to define
this virtual grid as kind of a meta grid. It is not really clear whether
a whole dune grid interface can be implemented. At least, traversal over
the leaf entities of the virtual grid can be implemented.

Best,
Simon


-------------- next part --------------
A non-text attachment was scrubbed...
Name: signature.asc
Type: application/pgp-signature
Size: 801 bytes
Desc: OpenPGP digital signature
URL: <https://lists.dune-project.org/pipermail/dune-functions/attachments/20190202/daef0ebe/attachment.sig>