  This paper is a computational bifurcation analysis of a non-linear partial
differential equation (PDE) characterizing equilibrium configurations in Micro
electromechanical Systems (MEMS). MEMS are engineering systems that utilize
electrostatic forces to actuate elastic surfaces. The potential equilibrium
states of MEMS are described by solutions of a singularly perturbed elliptic
nonlinear PDE. We develop a numerical method which couples a finite element
approximation with mesh refinement to a pseudo arc-length continuation
algorithm to numerically obtain bifurcation diagrams in the physically relevant
two dimensional scenario. Several geometries, including a unit disk, square,
and annulus, are studied to understand the behavior of the system over a range
of domains and parameter regimes. We find that solution multiplicity, and
importantly the potential for bistability in the system, depends sensitively on
the parameters. In the annulus domain, symmetry breaking bifurcations are
located and asymmetric solution branches are tracked. This work significantly
extends the envelope for numerical characterization of equilibrium states in
microscopic electrostatic contact problems relating to MEMS.
