[firedrake] Mesh coordinates for higher degree of FE

[Matthew Knepley]

On Wed, Dec 11, 2019 at 7:12 AM Luca Mechelli <luca.mechelli at uni-konstanz.de>
wrote:

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> Dear all,
>
> I am wondering if it is possible to access the list of all points used by
> piecewise quadratic (or higher order) elements on a regular mesh.
> For example, let's say I use the simple example of the Poisson equation at
> https://www.firedrakeproject.org/demos/poisson.py.html
>
> If I print mesh.coordinates.dat.data_ro I can see the verteces of the
> mesh, but not the middle points used to build the quadratic elements, then
> if I print u.dat.data_ro the array has not the same length of
> mesh.coordinates.dat.data_ro because the middle points nodes are missing
> from the second array. Is it possible to obtain the full list of
> coordinates used? If yes, how?
>

The problem is that this is a completely crazy way of thinking about finite
elements that we are stuck with thanks to Tom Hughes
and his followers. Coordinates on a mesh are a field, just like any other
field, and are discretized using a finite element space, just
like any other field. What are the consequences of this?

1) You should not assume that the dual basis for coordinates are point
evaluation functionals, or another way, not assume
     that we store coordinates by the values at certain points that you
know ahead of time. For example, we might store them
     as Bezier curves, or splines, or using model DG.

2) If you want the coordinates at a point, you should interpolate them
there just as you would a solution field.

3) You cannot assume that coordinates use the same space, or degree, as the
solution.

I am sure the FD folks can tell you the nicest way to interpolate.

  Thanks,

     Matt


> Thank you in advance,
> best regards,
> Luca Mechelli
>
>
> --
> Dr. Luca Mechelli
> Wissenschaftlicher Mitarbeiter
> Arbeitsgruppe Numerik
> Fachbereich Mathematik und Statistik
>
> Universität Konstanz
> Fach D 198
> D-78464 Konstanz
> Telefon: +49-(0) 7531-88-3646
>
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-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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