  Numerical and analytical methods are developed for the investigation of
contact sets in electrostatic-elastic deflections modeling micro-electro
mechanical systems. The model for the membrane deflection is a fourth-order
semi-linear partial differential equation and the contact events occur in this
system as finite time singularities. Primary research interest is in the
dependence of the contact set on model parameters and the geometry of the
domain. An adaptive numerical strategy is developed based on a moving mesh
partial differential equation to dynamically relocate a fixed number of mesh
points to increase density where the solution has fine scale detail,
particularly in the vicinity of forming singularities. To complement this
computational tool, a singular perturbation analysis is used to develop a
geometric theory for predicting the possible contact sets. The validity of
these two approaches are demonstrated with a variety of test cases.
