  We investigate the scaling of the ground state energy and optimal domain
patterns in thin ferromagnetic films with strong uniaxial anisotropy and the
easy axis perpendicular to the film plane. Starting from the full
three-dimensional micromagnetic model, we identify the critical scaling where
the transition from single domain to multidomain ground states such as bubble
or maze patterns occurs. Furthermore, we analyze the asymptotic behavior of the
energy in two regimes separated by a transition. In the single domain regime,
the energy $\Gamma$-converges towards a much simpler two-dimensional and local
model. In the second regime, we derive the scaling of the minimal energy and
deduce a scaling law for the typical domain size.
