This paper presents simulations of an atmospheric pressure air discharge in a point-to-plane geometry with a dielectric layer parallel to the cathode plane. Experimentally, a discharge reignition in the air gap below the dielectrics has been observed. With a 2D fluid model, it is shown that due to the fast rise of the high voltage applied and the sharp point used, a first positive spherical discharge forms around the point. Then this discharge propagates axially and impacts the dielectrics. As the first discharge starts spreading on the upper dielectric surface, in the second air gap with a low preionization density of 10^4~\textc\textm^-3 , the 2D fluid model predicts a rapid reignition of a positive discharge. As in experiments, the discharge reignition is much slower, a discussion on physical processes to be considered in the model to increase the reignition delay is presented. The limit case with no initial seed charges in the second air gap has been studied. First, we have calculated the time to release an electron from the cathode surface by thermionic and field emission processes for a work function φ ∈ ≤ft[3,4\right] eV and an amplification factor β ∈ ≤ft[100,220\right] . Then a 3D Monte Carlo model has been used to follow the dynamics of formation of an avalanche starting from a single electron emitted at the cathode. Due to the high electric field in the second air gap, we have shown that in a few nanoseconds, a Gaussian cloud of seed charges is formed at a small distance from the cathode plane. This Gaussian cloud has been used as the initial condition of the 2D fluid model in the second air gap. In this case, the propagation of a double headed discharge in the second air gap has been observed and the reignition delay is in rather good agreement with experiments.