Re: [Libmesh-users] Finite element types

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

On Tue, 30 Oct 2018, Hubert Weissmann wrote:

> On 30.10.2018 16:50, John Peterson wrote:
>>
>>  On Tue, Oct 30, 2018 at 9:28 AM Hubert Weissmann
>>  <hub...@gm... <mailto:hub...@gm...>> wrote:
>>
>>      Dear all,
>>
>>      I have trouble with the regularity of my solution (using
>>      Lagrange-elements, only continuity is ensured), and therefore
>>      thinking
>>      whether I should switch to Clough-Tocher elements or Hermite elements.
>> 
>>
>>  Are you solving a problem (e.g. biharmonic equation) where you expect the
>>  solution to be in C^1?
> 
> I forgot to mention: I am solving the laplace equation.
> In principle I expect my solution to be at least in C^2; so any improvement 
> of the continuity is appreciated.

To be fair, you're still getting the wrong answer with either element,
so you have room to play: If you wanted improved continuity out of
Lagrange, you can postprocess to get it.

But an advantage of the Hermites is that because you're *not*
postprocessing you basically "save DoFs" by not wasting them on
derivative discontinuities you expect not to exist.  So you end up
using ~8 DoFs per element for cubics in 3D instead of ~27 DoFs per,
and yet you don't lose much accuracy by it.  IMHO this sort of "k"
refinement that increases smoothness ought to go hand in hand with p
refinement, except that most software isn't set up to make that easy.
If you know you're never going to be applying non-smooth forcing
functions or boundary conditions then asking for more continuity
uniformly is still tempting.

> In principle, I agree with you; in the FE-region, it looks quite fine with 
> Lagrange elements. But since the boundary to infinite elements is really bad, 
> I hope to improve with other elements.
> The main disadvantage is that none of them are implemented for infinite 
> elements nor for Tets, which I use since they are much easier to setup; but I 
> might change this...

If you're mixing them with infinite elements, then the main
disadvantage I see is that it's going to be impossible for you to
handle the domain!  IIRC currently the HERMITE element type requires
your elements' xi/eta/zeta axes to line up everywhere with x/y/z (e.g.
you're forced to use rectangles in 2D), and even in principle the way
they handle mixed derivatives requires you to have continuous edges
going through nodes (e.g. you have to be working on a diffeomorphism
from a grid of squares); but that means you're going to have at least
4 square corners on the edges of your FE boundary, and that means the
InfFE approximation extending from those corners is going to be lousy,
isn't it?
---
Roy