Re: [Libmesh-users] Finite element types

[Stogner, R. H <roy...@ic...>]

On Tue, 30 Oct 2018, Hubert Weissmann wrote:

> I must admit that I didn't look closely at the current implementation; I
> just saw that the elements I would like to use are not done, so far.
> Is there a strict reason why the elements must align with the axes?

Alignment with the axes isn't strictly required, but alignment of
edges approaching from opposite sides of a node is.  So you could get
parallelograms or something marginally better, but you'll never have a
node of valence 3.

> I have smooth parameters and driving forces, but my elements are
> rather arbitrary and roughly align in spheres.

That's what I was afraid of.

> There are people who actually use InfFE with a 'rectangular'
> interface, but I would actually like to avoid it...
> Are the Clough-elements more promising?

Well, they let you use arbitrary domains in 2D, but the mid-edge
degrees of freedom cost you - if you subdivide quads to 2
Clough-Tocher triangles each, then Clough-Tocher gives you roughly 6
DoFs per quad, as opposed to 4 for Hermite or 9 for cubic Lagrange.

More importantly, now the integration itself is going to cost you:
each of those 2-macrotriangle quads has 6 subtriangles in it, and if
you use the simple clough-tocher quadrature I did of slapping Gaussian
points on each subtriangle then you're doing roughly 10 times as many
point evaluations.  This can still be worth it if the alternative
would have involved a mixed method (18 DoFs for cubic Lagrange and
more complexity at every quadrature point) but I don't think it's
worth it just because you have a smooth solution.

How big are your elements?  Can you just use fewer elements with
higher p (Hierarchics)?  That will still leave you with lower
smoothness so if you directly need smoothness it's not an option but
if you're just trying to reduce the problem size for a given accuracy
then that'd be the first thing I'd try.
---
Roy