Moving mesh simulation of contact sets in two dimensional models of elastic–electrostatic deflection problems

•New moving mesh method with application to Micro Electro Mechanical Systems (MEMS).•Resolution of singular solutions in fourth order parabolic PDEs.•Robust handling of non-convex and non-simply connected two dimensional domains. 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.