I’m modeling a superconductor using the London equation ∇^2 A - A/λ^2 = 0, where A is the vector potential (a vector in 2D or 3D space) and λ is a constant. In the superconductor λ ≠ 0, while outside it λ = 0. In the usual way, my simulation space Ω is bounded by a large outer boundary ∂Ω on which A = Aouter(x, y, z). The superconductor (some shape such as a plate or a sphere) is then a smaller region η with boundary ∂η inside ∂Ω.

The issue is that there is also a boundary condition on ∂η, namely, that n·A = 0 on η. So there are two things I need to be able to do:

- Enforce a boundary condition on a surface that is inside the overall volume; both the inside and outside of this surface are included in the analysis.
- This boundary condition is that n·A = 0. In other words, it does not involve the derivatives of A, but rather the value of A (or, really, one of its components).

Any suggestions?