  Starting from the Ginzburg--Landau model in a planar simply connected domain,
with a local compactly supported applied magnetic field, we derive an effective
model in the strong field limit, defined on a non-simply connected domain. The
effective model features oscillations in the Little-Parks and Aharonov--Bohm
spirit. We discuss also a similar question for the lowest eigenvalue of the
magnetic Laplacian.
