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We use conformal maps to study a free boundary problem for a two-fluidelectromechanical system, where the interface between the fluids is determinedby the combined effects of electrostatic forces, gravity and surface tension.The free boundary in our system develops sharp corners/singularities in certainparameter regimes, and this is an impediment to using existing "single-scale"numerical conformal mapping methods. The difficulty is due to the phenomenon ofcrowding, i.e. the tendency of nodes in the preimage plane to concentrate nearthe sharp regions of the boundary, leaving the smooth regions of the boundarypoorly resolved. A natural idea is to exploit the scale separation between thesharp regions and smooth regions to solve for each region separately, and thenstitch the solutions together. However, this is not straightforward asconformal maps are rigid "global" objects, and it is not obvious how one wouldpatch 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 carryout this program of stitching conformal maps on different scales together. Wesuccessfully apply our method to the electromechanical model problem anddiscuss how it generalizes to other situations.

Sharp interfaces in two dimensional free boundary problems: Efficient interface calculation via matched conformal maps