Assembly with complex numbers

[gatling-nrl]

I'm trying to implement a form for $\int uv \ dx$ where u and v are complex.

Naively, I write $\int (u_{re} + iu_{im})(v_{re}+iv_{im}) \ dx$

which leads me to some bilinear form like $i\int u_{re} v_{im} \ dx$

Now I imagine code like

real_basis = skfem.CellBasis(mesh, element)  #P1 should suffice
imag_basis = skfem.CellBasis(mesh, element)  #P1 should suffice

@skfem.BilinearForm
def form(u, v, _):
    return u * v  # <-- this can't be right....
   # how to deal with those terms that need u from real and v from imag?

form.assemble(....?

How do i accomplish this?

EDIT:

It seems like I'm thinking about it all wrong. In the Poisson equation, for example, it seems that the unknown (complex) potential is modelled by real basis functions? I.e.

$u = \sum_n a_n \varphi_n$

the $\varphi_n$ basis functions are real, and only the $a_n$ are complex? In which case I would only need the one real_basis above?

[kinnala]

Perhaps this helps:

scikit-fem/tests/test_assembly.py

Line 432 in 6159aa3

class TestComplexValuedAssembly(TestCase):

I don't know if it was used anywhere.