  Streamer discharges play a central role in electric breakdown of matter in
pulsed electric fields, both in nature and in technology. Reliable and fast
computations of the minimal model for negative streamers in simple gases like
nitrogen have recently been developed. However, photoionization was not
included; it is important in air and poses a major numerical challenge. We here
introduce a fast and reliable method to include photoionization into our
numerical scheme with adaptice grids, and we discuss its importance for
negative streamers. In particular, we identify different propagation regimes
where photoionization does or does not play a role.
