Hi,
The degree 1 Nédélec element (first kind) maintains continuity for tangential electric fields Et but not for the normal component of flux density Dn. Is there an element that ensures continuity for both the tangential component of E and the normal component of D at edges?
What should the behavior across the element facet (different to an edge in 3D be)?
I would suggest consulting DefElement: Categories
as it lists most elements.
Maybe @mscroggs has an idea.
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Thank you. I am currently working on a 2D electromagnetic problem, and by “edges”, I was referring to facets. My goal is to ensure the continuity of the normal component, D_n = \epsilon E_n at element boundaries, either by selecting an appropriate vector-valued element or by incorporating the constraint \nabla \cdot D = \nabla \cdot (\epsilon E)=0 into the problem. Is there a way to introduce such a constraint into the following weak form?
\int_\Omega (\nabla \times E \cdot \nabla \times W) -k_0^2 \epsilon_r E\cdot W d\Omega +jk_0 \int_\Gamma \sqrt\epsilon_r \, ( n\times E)\cdot( n\times W) d\Gamma = j\omega\mu_0 \int_\Omega J\cdot W d\Omega
If both the tangential component and normal component should be continuous, you can use vector Lagrange elements. It will not be divergence free though. Another way of enforcing this would be to use Nedelec elements and a weak enforcement of the normal divergence.
Thank you very much. Could you please explain further what is meant by ‘weak enforcement of the normal divergence’? I want to emphasize that E is not divergence-free, and its normal component exhibits a discontinuity. The condition of being divergence-free holds for D = \epsilon E.
See for instance the intro of https://www.sciencedirect.com/science/article/pii/S0045782523006175
so I would look for: Google Scholar
I would probably read: https://epubs.siam.org/doi/abs/10.1137/19M1290966
Maybe @jpdean can give some further hints, as this is his cup of tea.