How can one compute the Fichera norm of a function g \in H^{1/2}(\partial \Omega)? That is, the quantity
||g||_{1/2, \partial \Omega}^2 := \int_{\partial \Omega} \int_{\partial \Omega} \frac{|g(x)-g(y)|^2}{|x - y|^2} ds(x) ds(y).
I think this might not be possible with dolfinx since ufl documentation says it can deal with this:
a(\mathbf{v}; \mathbf{w})
=
\sum_{k=1}^{n_c} \int_{\Omega_k}
I^c_k(\mathbf{v}; \mathbf{w}) dx
+ \sum_{k=1}^{n_e} \int_{\partial\Omega_k}
I^e_k(\mathbf{v}; \mathbf{w}) ds
+ \sum_{k=1}^{n_i} \int_{\Gamma_k}
I^i_k(\mathbf{v}; \mathbf{w}) dS.
Don’t you think, it would be possible to first integrate over one boundary and then over the other?
Using stock Fenicsx UFL, this is not possible. I am also interested in similar types of boundary integrals (Fredholm types I and II), but at present there seems to be no straightforward way to do this. BEMPP was one way that you approach boundary integral problems with legacy Fenics, but I am not sure if they have a version that works with Fenicsx.
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BEMPP works with FEniCSx, see for instance: Jupyter Notebook Viewer
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Apparently, there is Fenicsx support for BEMPP. However, BEMPP only considers 2D surfaces in 3D, while my boundaries are in 1D, since I am solving 2D problems.
Did you only consider the 3D case so far?
Yes, I have only been working with 3-D problems.