  We use conformal maps to study a free boundary problem for a two-fluid
electromechanical system, where the interface between the fluids is determined
by the combined effects of electrostatic forces, gravity and surface tension.
The free boundary in our system develops sharp corners/singularities in certain
parameter regimes, and this is an impediment to using existing "single-scale"
numerical conformal mapping methods. The difficulty is due to the phenomenon of
crowding, i.e. the tendency of nodes in the preimage plane to concentrate near
the sharp regions of the boundary, leaving the smooth regions of the boundary
poorly resolved. A natural idea is to exploit the scale separation between the
sharp regions and smooth regions to solve for each region separately, and then
stitch the solutions together. However, this is not straightforward as
conformal maps are rigid "global" objects, and it is not obvious how one would
patch two conformal maps together to obtain a new conformal map. We develop a
"multi-scale" (i.e. adaptive) conformal mapping method that allows us to carry
out this program of stitching conformal maps on different scales together. We
successfully apply our method to the electromechanical model problem and
discuss how it generalizes to other situations.
