Dear Floriane,

I think you can do this in a manner analogous to the way that Thomasz did fluid-structure interaction.

We define a mixed function space with two continuous components: (V_d, V_p). We’re going to use the first component for the solution in the disk and the second component for the solution outside the disk.

We further define two DG0 functions I_d and I_p such that I_d is 1 inside the disk and 0 outside, and I_p = 1 – I_d.

You can now write the Laplace equation for the two parts using essentially normal Firedrake code except that you multiply the test function by the appropriate indicator function. This ensures that you only actually assemble integral contributions on the correct part of the domain.

Now we just need the surface integral over the facets on the edge of the disk. Observe that this is the only place in the domain where both indicator functions are positive. You can write your jump integrals using the  *dS measure. You can restrict the intervals to the relevant edges by multiplying the integrand by 4*avg(I_p)*avg(I_d).

If

W = V_d * V_p

and

u_d, u_p = TrialFunctions(W)

then you can write the jump as:

avg(u_d) – avg(u_p)

The avg is mathematically unnecessary and does nothing, however in UFL all terminals that appear in dS integrals have to be restricted (you could just as well write u_d(‘+’) – u_p(‘+’)).

Finally, when you come to the solve, you need to ensure that all the V_d nodes outside the disk are eliminated, and all the V_p nodes inside the disk too. You do this using the same DirichletBC subclassing trick which is in https://www.firedrakeproject.org/demos/linear_fluid_structure_interaction.py.html

I realise that is slightly involved, but it should work and the syntax will be fairly clean once you have done it. Feel free to ask for clarifications.

Regards,

David