Re: [deal.II] Locally divergence free DG basis

[Wolfgang Bangerth]

On 3/13/25 06:55, Florian Schulze wrote:
I am trying to solve the magnetohydrodynamics (MHD) equations using the 
discontinuous Galerkin method with deal.II.
Currently, I face the problem of the divergence free condition of the 
magnetic field, ∇⋅𝐁 = 0.
One of the ways I want to address this problem is using locally 
divergence free elements,
i.e. an FE basis 𝛟(𝐱) that fulfils ∇⋅𝛟(𝐱) = 0 on the cell.
(Face jumps of the normal component can still introduce divergence 
globally - I plan to use different methods to deal with that.)

I found that this topic was already discussed in the forum before:

  * Locally divergence free FE space <https://groups.google.com/g/
    dealii/c/uGY2ltkMJ-E/m/6KzZm1u6l-EJ>
  * Custom shapeset questions <https://groups.google.com/g/dealii/c/
    NDiUu3qESOM/m/magIbF_FCQAJ>

But I don’t see that it made it into deal.II.
Am I missing something? Does deal.II have a divergence free FE basis?
Or does someone already have/know of an implementation?

If not, what would be the best way to go about implementing it?
After a short look in the documentation, seems like implementing a new 
TensorPolynomial <https://www.dealii.org/current/doxygen/deal.II/ 
classTensorPolynomialsBase.html> and using with in the FE_DGVector 
<https://www.dealii.org/current/doxygen/deal.II/classFE__DGVector.html> 
is a possibility?

Florian,
none of the previous work was ever proposed for integration into 
deal.II, so there are no examples. But in general, implementing finite 
elements is not "terribly" difficult as it is quite a localized problem: 
You really only have to understand how the FiniteElement class works and 
its interface. The biggest challenge is perhaps to understand the many 
intermediate classes we have come up with over the years that abstract 
the problem to one where you provide a class that describes the 
polynomials and then everything else works automatically.

If your element is indeed discontinuous, then you are right that 
FE_DGVector is the right starting point. In that case, you also don't 
have to worry about one of the more labor-intensive parts, namely 
dealing with hanging node constraints.

Obviously, we'd be excited to integrate whatever you come up with. 
Please do ask questions if you get stuck!

Best
 W.

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