  Aiming at the study of critical phenomena in the presence of boundaries with
a non-trivial shape we discuss how lattices with an adaptive lattice spacing
can be implemented. Since the parameters of the Hamiltonian transform
non-trivially under changes of the length-scale, adapting the lattice spacing
is much more difficult than in the case of the numerical solution of partial
differential equations, where this method is common practice. Here we shall
focus on the universality class of the three-dimensional Ising model. Our
starting point is the improved Blume-Capel model on the simple cubic lattice.
In our approach, the system is composed of sectors with lattice spacing a, 2 a,
4 a, ... . We work out how parts of the lattice with lattice spacing a and 2 a,
respectively, can be coupled in a consistent way. Here, we restrict ourself to
the case, where the boundary between the sectors is perpendicular to one of the
lattice-axis. Based on the theory of defect planes one expects that it is
sufficient to tune the coupling between these two regions. To this end we
perform a finite size scaling study. However first numerical results show that
slowly decaying corrections remain. It turns out that these corrections can be
removed by adjusting the strength of the couplings within the boundary layers.
As benchmark, we simulate films with strongly symmetry breaking boundary
conditions. We determine the magnetization profile and the thermodynamic
Casimir force. For our largest thickness L_0=64.5, we find that results
obtained for the homogeneous system are nicely reproduced.
