Compute the face embedding matrix for FE_NedelecSZ

[Sebastian Kinnewig]

Hello all,

I am currently working on an adaptive solver for the time-harmonic Maxwell equation. At the moment I am using the Nédélec elements provided by the class FE_Nedelec. Since FE_Nedelec requires a consistent orientation of the faces, I would like to switch to FE_NedelecSZ (therefore I already implemented the hessian in FE_NedelecSZ #12771).

To make the switch to FE_NedelecSZ I need to compute the face embedding. I tried to compute the face embedding via the function FETools::compute_face_embedding_matricies() and pass the resulting face embedding matrix to the interface_constraints (similar to what is done in the FE_Nedelec class). But that doesn't seem to work, does anyone have any ideas or suggestions as to what the problem could be?

Thank you in advance!

Best Regards,

Sebastian

[Wolfgang Bangerth]

On 9/21/21 8:35 AM, Sebastian Kinnewig wrote:
>
> To make the switch to FE_NedelecSZ I need to compute the face embedding.
> I tried to compute the face embedding via the function
> FETools::compute_face_embedding_matricies() and pass the resulting face
> embedding matrix to the interface_constraints (similar to what is done
> in the FE_Nedelec class). But that doesn't seem to work, does anyone
> have any ideas or suggestions as to what the problem could be?

Can you describe what happens (i.e., what "that doesn't seem to work"
actually means)?

Best
W.


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

[Sebastian Kinnewig]

Thank you for the first answer!

When I implement the computation of the embedding matrix , as done in FE_Nedelec, in FE_NedelecSZ the program compiles and runs without any warnings or errors (even in debug mode).

But from the results it appears that there still is a problem with the hanging nodes.

Maybe it is the best when I attach some pictures of the solution. For the test case I computed a reference solution to which I know the analytical solution.
All faces in that test case are oriented (therefore there should be no sign conflict) moreover, it works when I use FE_Nedelec (see appendix). With just global refinement also FE_NedelecSZ works fine.

So my best guess is, that there is still a problem with the hanging nodes. (Are there maybe more functions that need to be called for the hanging nodes to work correctly? At the moment I am only define this->interface_constraints in FE_NedelecSZ.)

Best,
Sebastian

 

[Alexander]

Hi Sebastian

FE_NedelecSZ does not support hanging nodes, i.e. the mesh must be conforming.

Implementing handing-node constraints for this element is possible, but would require a special approach since constraints are not static. If you feel like implementing this, I would be glad to discuss -- I happen to miss this functionality too, but I've never come to a point where I can devote enough time to this.

Best

Alexander

[Wolfgang Bangerth]

On 9/22/21 2:37 AM, Sebastian Kinnewig wrote:
>
> When I implement the computation of the embedding matrix , as done in
> FE_Nedelec, in FE_NedelecSZ the program compiles and runs without any
> warnings or errors (even in debug mode).
> But from the results it appears that there still is a problem with the
> hanging nodes.

Alexander knows this element much better than I do, so I have little to
add. The current way we deal with constraints lives by the assumption
that on every cell, the shape functions are mapped from a reference cell
in the same way. I believe that that is not actually the case for the SZ
element because, if I recall correctly, the sign (vector direction) of
shape functions depends on the node numbers. A check for you might be to
see whether at hanging nodes the *magnitude* of the solution is correct
but the *sign* is wrong.