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.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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