DOF ordering H(div) elements

[mbr]

Hello,

I’m currently trying to figure out, how the degrees of freedom (DOF) are ordered for H(div) conforming elements (e.g. Raviart-Thomas elements). For Lagrangian elements this is quite simple as one can tabulate the physical DOF coordinates, which is not intended for the H(div) space.

I looked up the definition of the Raviart-Thomas elements under

DefElement: Raviart–Thomas

If we assume, we have an element as depicted below, is then the ordering of the DOF (green arrows) of the RT_0 (in fenics nomenclature RT-Space with oder 1) correct? Or In other words: Are I’m correct that the ordering of the DOFs is done in fenicsX based on the ordering of the nodes of the mesh element?

Just some further information on the definition of the above depicted element: I construct a mesh, read out the node ordering from mesh.geometry.dofmap and the associated nodal coordinates by extracting the - in this case - 0, 3, 2 row of the array mesh.geometry.x.

Kind regards,
Maximilian

[dokken]

@mscroggs has all the details regarding this.

I’ll leave a brief summary here:

• The input mesh ordering is not kept, see: Quad mesh nodes reordered upon reading XDMF file - #2 by dokken
• As mentioned in this post, DOLFINx uses dof-permutations to make integrals map back to the reference element, see: [2102.11901] Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes
  for details.

[mscroggs]

The ordering of the DOFs in FEniCSx on the reference cell follows the same order as shown on DefElement. But as @dokken says, this is complicated by the fact the mesh nodes are reordered when meshes are created, so the numbering of a cell in a mesh might not match the input.