[Dune] Dune::HierarchicalOrthogonalShapeFunctionSet< C, T, d > Class Template Reference

[Tatiana Voitovich]

Thank you very much for your answer!

>> could you please let me know, on what it depends, which version --- svn
>> or
>> release to use
>> for implementing a certain numerical scheme in DUNE+DUNE-fem.
>
> Please look through the recent changes, whether there are any
> features, whihc require you to use the development branch.

As far as I have understood, I will need to use the new package dune-disc.
To be precise, I will have to write a class, perhaps useful for many users
of dune, alternative to

Dune::MonomialShapeFunctionSet< C, T, d > Class Template Reference

which could have the name

Dune::HierarchicalOrthogonalShapeFunctionSet< C, T, d > Class Template
Reference,

or

Dune::GeneralizedTensorProductShapeFunctionSet< C, T, d > Class Template
Reference

which will be a dune implementation of an hierarchical orthogonal basis
functions set introduced independently by  Proriol, Karlin and McGregor,
Koornwinder and Dubiner and
traditionally employed by $hp-$ spectral finite element methods, you could
have a look in
T. Warburton, An explicit construction of interpolation nodes on the
simplex, J. Eng. Math. (2006),
247-262.
The other matter, in addition to the orthogonality property, why we would
like to use this set, is that those basis functions contain in their
expressions Jacobi polynomials, therefore
the diagonal mass matrix is very good conditioned, while the (cartesian
coordinates) monomials
function set is nearly linear dependent for p results in a mass matrix
that is nearly singular starting from the degree of interpolation
polynomials $p \gtrsim 7$ (see, f.e. M. Dubiner, Spectral methods on
triangles and other domains, 1991).

As far as I know, deal II, a structured-grid-branch, employes this basis
set extensively.

For this shape function set I have the elemental operations (such as
quadrature approximation
of the mass matrix) programmed in C.

With best regards,
Tatiana