[Dune] reference simplex in higher space dimensions, rotational symmetry of the quadrature points?

[Martin Nolte]

Dear Tatiana,

as all n-dimensional simplices are identical up to an affine transformation, 
you can simply implement the mapping from your favorite reference simplex to 
the DUNE reference simplex. This allows you to switch between the reference 
simplices. For example, if you have a quadrature for your favorite reference 
simplex, you can transform it into a quadrature on the DUNE reference simplex 
using the mapping you created above (by simply mapping the quadrature points, 
the weights remain the same). Of course, you can port anything you like in 
this way.

Cheers,

Martin

Tatiana Voitovich wrote:
> Hi Martin,
> 
> I am writing to you with respect to you contribution to the Dune
> development creating the code for a reference simplex of higher
> dimensions. To our believe, the task of development of a code that would
> by simple recompilation run for a higher space dimension is without any
> doubt a fascinating goal.  But as the most of the simulation programs
> should work for the dimensions 2 and 3, we could perhaps make a suggestion
> that it would be fine to have a possibility to use the reference of a
> special form for the dimensions 2 and 3. For example, for our
> implementation of LDG schemes for nonlinear degenerate
> convection-diffusion  equations in 2D we have choosen the equilateral
> triangle with the side 2 centered at the origin, because we have learned
> from publications, in particular from
> J. H. Hesthaven, C.-H. Teng, Stable  pectral methods on tetrahedral
> elements, SIAM J. Sci. Comp., 21 (2000) 2352-2380 that such reference
> elements allow constriction of quadrature points posessing a certain type
> of quasi-rotational symmetry (in the Cartesian coordinates) that is
> reported in this publication as be "important both for the quality of the
> interpolation and for the construction of efficient discrete
> differentiation". At the moment, however, we ourselves are using
> "asymmetric" points in that simplex, corresponding to a collapsed
> coordinate transform.
> 
> If I am not right in something that I have written, please let me know that.
> 
> Wishing you a success in you work,
> 
> with best regards,
> Tatiana
> 
> 

-- 
Martin Nolte <nolte at mathematik.uni-freiburg.de>

Universität Freiburg                                   phone: +49-761-203-5642
Abteilung für angewandte Mathematik                    fax:   +49-761-203-5632
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