The present approach to modeling and design of microelectronic and microelectromechanical devices (MEMS) (hereafter simply referred to as a microdevice) involves the generation of a geometric model for the complicated two or three-dimensional microdevice, the generation of a mesh for the geometric model, a mesh based numerical analysis, and postprocessing steps such as visualization. The time consuming steps in such an approach are the generation of a mesh and mesh-based numerical analysis. Meshless methods, which do not require the generation of a mesh, are very attractive for numerical solution of partial differential equations. In this paper we introduce a new meshless technique, referred to as a point collocation method, for numerical solution of partial differential equations. The effectiveness of the point collocation method is demonstrated by solving the one and two-dimensional Poisson equation, the solution of which is required in the analysis of both electronic and microelectromechanical devices.