This paper presents the application of the ghost fluid method (GFM) to solve Poisson's equation for streamer discharge simulations between electrodes of complex geometries. This approach allows one to use a simple rectilinear grid and nevertheless take into account the influence of the exact shape of the electrode on the calculation of the potential and the electric field. First, the validity of the GFM approach concerning the computation of the electric field is demonstrated by performing direct comparisons in a point-to-plane geometry of the Laplacian potential and electric field calculated with this method and given by the analytical solution. Second, the GFM is applied to the simulation of a positive streamer propagation in a hyperboloid-to-plane configuration studied by Kulikovsky (1998 Phys. Rev. E 57 7066–74). Very good agreement is obtained with the results of Kulikovsky (1998) on all positive streamer characteristics during its propagation in the interelectrode gap. Then the GFM is applied to simulate the discharge in preheated air at atmospheric pressure in point-to-point geometry. The propagation of positive and negative streamers from both point electrodes is observed. After the interaction of both discharges, the very rapid propagation of the positive streamer towards the cathode in the volume pre-ionized by the negative streamer is presented. This structure of the discharge is in qualitative agreement with the experiment.