Is barycenter refinement for simplices implemented?

[Kyle Schwiebert]

I'm sorry to ask such a basic question here, as it feels like I should be able to answer this for myself, but having a look at some relevant git and source code, I could not find the answer. I also could not find anything in the documentation and it seems the question is also unaddressed on this forum:

How does global refinement work in the case of simplices, and in particular, is it barycenter refinement? I suspect it would not be, since iterated barycenter refinement eventually produces a very bad mesh. Is if it is not the default, is it implemented at all? Or is this a case of one being better off importing a mesh from another program?

Nonetheless, barycenter refinement seems crucial for using Scott-Vogelius finite element spaces in either 2D or 3D. All the other necessary components (FE_SimplexP and FE-SimplexDGP are already present).

[Wolfgang Bangerth]

Kyle:

> I'm sorry to ask such a basic question here, as it feels like I should be able
> to answer this for myself, but having a look at some relevant git and source
> code, I could not find the answer. I also could not find anything in the
> documentation and it seems the question is also unaddressed on this forum:
>
> How does global refinement work in the case of simplices, and in particular,
> is it barycenter refinement? I suspect it would not be, since iterated
> barycenter refinement eventually produces a very bad mesh. Is if it is not the
> default, is it implemented at all? Or is this a case of one being better off
> importing a mesh from another program?

As you already say, barycenter refinement has the disadvantage that it leads
to degenerate elements if you do it multiple times. As a consequence, we use
for triangles what we already use for quadrilaterals: We subdivide each
triangle into four children using the edge midpoints -- this actually leads to
four children congruent with its parent, so there is no degeneration at all.
It leads to hanging nodes, but we know how to deal with those.

> Nonetheless, barycenter refinement seems crucial for using Scott-Vogelius
> finite element spaces in either 2D or 3D.

This is the wrong perspective. If barycentric refinement is necessary, then
you just start with a good mesh (like the one generated with the algorithm
above) and you *consider* each cell subdivided into three *for the purposes of
definition of the element*. (Like in Fig. 1 of
http://www.math.clemson.edu/~vjervin/papers/erv104.pdf, for example: The mesh
on the right does not exist as a set of cells; it is only used to define shape
functions on the mesh on the left.) This is the same construction as we use in
the FE_Q_iso_Q1 element, for example.

Another element that is generally created like this is the
(Hsieh-)Clough-Tocher element. See for example p. 117 of
https://people.sc.fsu.edu/~jburkardt/classes/fem_2011/chapter6.pdf and here:
https://www.math.utah.edu/~pa/MDS/felm.html

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/