the set of permissible complex forms is overly restrictive

[danshapero]

The current implementation of complex forms makes strong assumptions about differentiability w.r.t. the solution u. These assumptions forbid certain problems like the Ginzburg-Landau equation. The Ginzburg-Landau equation is most naturally defined as the Euler-Lagrange equations for the action functional

E = (0.5 * inner(grad(u), grad(u)) + 0.25 * a * (1 - inner(u, u)) ** 2) * dx

The problem here is the inner(u, u) which is really the product of u with its complex conjugate.

I think there's a principled solution you get by taking a page out of the several complex variables textbook. You have to think of u and its complex conjugate as separate, independent variables; see the Dolbeault operators. To get a (k + 1)-form from a k-form, you differentiate w.r.t. either u or conj(u) depending on k. I think this does the right thing for all the interesting variational problems for complex-valued fields that I know of but I don't have a good reference.

If this issue is worth solving, it would be good to come up with a list of other problems where the feature would be useful as candidate demos. The other two that I know of are Gross-Pitaevskii and nonlinear Schrödinger. There's probably something in solid-state physics or semiconductor physics, I just don't know what it is.

@pbrubeck @wence- please correct or add anything I missed from our conversation.

[wence-]

To get a $(k + 1)$-form from a $k$-form, you differentiate w.r.t. either $u$ or $\text{conj}(u)$ depending on $k$.

Just to try and clarify here. The $k$-form you mean in this sentence is, by my reading of the linked page, a differential $k$-form (i.e. a part of some complex). And then the Dolbeault operators give you a way to define the exterior derivative.

But this doesn't obviously allow me to compute the directional derivative of (say) a linear (in the sense of only has a test function) form to produce a bilinear form (has a test and trial function). Because that's not the same thing as attempting to apply the exterior derivative to map from a differential-1-form to a differential-2-form, no?

[danshapero]

Just to try and clarify here. The k -form you mean in this sentence is, by my reading of the linked page, a differential k -form (i.e. a part of some complex). And then the Dolbeault operators give you a way to define the exterior derivative.

Woops I wrote that badly, what I meant was a firedrake.Form that takes k arguments. I was trying to summarize something Pablo said about how you should be able to take the derivative of the 0-argument form inner(u, u) * dx twice to get a mass matrix (strictly speaking it's off by a factor of two). Likewise for inner(grad(u), grad(u)). I think the first time you differentiate w.r.t. the conjugate of u to get a 1-argument form that is linear in the conjugate of the test function. The second time you differentiate w.r.t. u. Assuming I did all that right, you get a Hermitian matrix.

@jrmaddison mentioned the Wirtinger derivatives in slack, that's a better reference than the Dolbeault operators.