On Fri, 19 Sep 2014, Eduardo Hernandez wrote:
I used LAGRANGE finite elements, which I understand guarantee
continuity of the (eigen)functions across elements, but not of their
derivatives (C0 continuity).
That's right.
I also understand that in order to
enforce continuity of the 1st derivative I should switch to finite
elements based on Hermite interpolating polynomials, which I have
done with the following lines of code:
MeshTools::Generation::build_line( mesh, npoints, -L0, L0, EDGE3 );
…
eigen_system.add_variable( "p", THIRD, HERMITE );
and if I am not mistaken, this will ensure that the eigenvectors have C1
continuity (though please correct me if wrong).
You're 99% right. If you use an adapted mesh in 2D or higher things
get slightly trickier; you'd need to avoid spurious eigenvectors and
to enforce_constraints_exactly() on the eigenvectors the solver gives
you.
My question is: is it possible to enforce C2 continuity (continuity
of function, first and second derivatives)? I naively assumed that
this would be possible by using a higher order of HERMITE (e.g.
FIFTH) in the appropriate line above, but this does not appear to
work.
No; the higher order HERMITE elements give better approximation
accuracy but are still only C1 continuous.
If you need to add a C2 Hermite-like element yourself, we'd love to
accept a patch. Getting vector projections correct in a C2 space
would be complicated, but just coding up an element that works on
unadapted rectilinear grids would be straightforward, and the 1D
version would be easy.
---
Roy
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