This paper is devoted to the theoretical investigation of the breakdown in short discharge gaps of different geometries under the influence of the plasma stream from an external source. The structural feature is the presence of dielectric elements in the electric discharge gap area, which have high emission activity and the ability to accumulate a surface charge. The simulation was performed for a 2-D planar geometry of the discharge gap between two metal electrodes with dielectric coating surrounded by the gaseous medium. Charged particles generation and dynamics have been described by a system of partial differential equations in the diffusion-drift approximation within the two-fluid hydrodynamics plasma model. The basic advantages of the proposed model (such as a wide variety of boundary conditions and geometries of the discharge gap, the scalability of the critical parameters of the environment, and a simple representation of surface reactions) are demonstrated successfully. During computations, the range of gas pressure and the external preionization level at which the probability of low-voltage self-sustained discharge is high enough were identified.