Bulk-surface coupling with vector elements

[haddel]

Dear FEniCS community,

I am currently exploring bulk–surface coupling with vector finite elements (H(curl), H(div)) in Fenics/FEniCSx. In the simplest setting this corresponds to a curl–curl problem in the bulk coupled to a surface curl–curl problem on the boundary. From what I can tell, many FEM platforms do not even natively support the simpler case of a coupled Laplace–Beltrami problem, but a tutorial by J. S. Dokken suggested that Fenicsx could be a good fit for multi-dimensional coupling, which gave me hope.

For my problem I need to implement the tangential trace spaces H(curl_Γ) and H(div_Γ) on a curved two-dimensional surface Γ embedded in three dimensions. I would be excited to pursue this if possible and would be glad to share my work as a future contribution to the project, if this interests you.

At the moment, I only have a rough understanding of the internal structure of the multiple Fenics projects that comprise the finite element library. I would greatly appreciate any guidance, pointers, or discussion about this idea, including feasibility, suggested development paths, and any related ongoing work.

Best Julian

[dokken]

Just to clarify, you would like to implement a H-curl space on \Gamma which is the conforming boundary of a 3D domain \Omega? I don’t see any obvious issues with this.
The only thing that comes to mind is to be careful about global orientations, ref this length post:

Does Dolfinx's solver have an issue or require some preconditioning when handling certain meshes like the Icosahedral mesh?

I am going to try to illustrate the mistake I mean exists in the legacy code with a small example: from dolfin import * class OutwardNormal(UserExpression): def __init__(self, mesh, *arg, **kwargs): super().__init__(*arg, **kwargs) self.mesh = mesh def eval_cell(self, values, x, cell): c = Cell(self.mesh, cell.index) normal = c.cell_normal() scale = 1 if c.orientation == 0 else -1 values[0] = normal[0] * scale values[1] = normal…

with code at: Add manifold example with real spaces · Issue #118 · scientificcomputing/scifem · GitHub