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

[Tatiana Voitovich]

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