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

Oliver Sander sander at igpm.rwth-aachen.de
Thu Apr 2 14:28:42 CEST 2015


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.

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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