[dune-functions] New example for Taylor-Hood elements

[Carsten Gräser]

Am 03.04.2015 um 22:58 schrieb Carsten Gräser:
> Hi Oliver,
> 
> Am 02.04.2015 um 14:28 schrieb Oliver Sander:
>> Hi guys,
>>
>> I just added a second example file to our /examples directory.  This one is intended
>> to show how function space bases with non-trivial tree are used.  The code discretizes
>> the Stokes equation with a Taylor-Hood basis.  The setting is the well-known driven-
>> cavity problem: a unit square, no source term, homogeneneous Dirichlet boundary
>> conditions on three sides of the square, constant (non-zero) Dirichlet bc on the forth side.
>>
>> Have a look at the example if you find the time.  If you think that I did anything in
>> a suboptimal way feel free to push patches.
> in the local assembler you compute the local matrix indices manually.
> I would say that one should use localIndex() here. Unfortunately this
> will not work with the current implementation because the localIndex()
> method of the PQk basis is hard wired assuming that leaf==root.
> 
> In principle one could solve this by storing the indices in the leaf
> node. Then one could overwrite them if leaf!=root. Since we need this
> in any basis, we should have something like setLocalIndex() in
> GridFunctionSpaceBasisLeafNodeInterface.
I'm just trying to implement this. Unfortunately it will requires
to make copies in the PowerNode. Does anyone have a better idea?

Best,
Carsten
> 
> But there's a problem: If we use the node as leaf in a nontrivial
> tree, like Taylor-Hood does, this may lead to inconsistencies: We
> manipulate the local indices manually but still expect the LocalIndexSet
> to behave as it did before. I'm not sure if this is a good idea.
> We could solve this problem by requiring that LocalIndexSet does not
> use the localIndex() method of the nodes.
> 
> Best,
> Carsten
> 
>>
>> Unfortunately, when writing the example I found several situations where we are still
>> lacking infrastructure:
>>
>> - Apparently there is no solver for such a problem in dune-istl.  I'd volunteer to add
>>   such a solver, but I am unsure exactly what type of solver to implement.  A preconditioned
>>   Uzawa?  Please advice.
>> - While the TaylorHoodBasis class is sufficient to assemble the algebraic problem,
>>   it is not sufficient for pre- and post-processing.  How do you get the velocity
>>   part of a solution into a vtk file?  Currently, the code instantiates a separate
>>   PQ2NodalBasis object for this, but what we really want is something like 'view objects'
>>   on given bases, that allow to extract subtrees and the corresponding coefficients.
>> - Same for interpolation: To find the Dirichlet dofs I need to interpolate the identity
>>   function on the velocity basis.  This is currently not possible.
>>
>> Finally, there are some issues regarding the linear algebra.
>>
>> - The matrix data structure is now Matrix<BCRSMatrix<FieldMatrix<double,1,1> > >.
>>   This still ignores quite a bit of structure of the occupation pattern.  This is
>>   unfortunate, because we claim that dune-functions and in particular dune-istl
>>   is all about using this structure.
>> - The type for coefficient vectors is FieldVector<BlockVector<FieldVector> >.
>>   Strictly speaking this is against the rules, as FieldVectors are supposed
>>   to end recursion.
>>
>> Cheers,
>> Oliver

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