A system very similar to a dielectric barrier discharge, but with a simple stationary dc voltage, can be realized by sandwiching a gas discharge and a high-ohmic semiconductor layer between two planar electrodes. In experiments this system forms spatiotemporal and temporal patterns spontaneously, quite similarly to, e.g., Rayleigh-Benard convection. Here it is modeled with a simple discharge model with space charge effects, and the semiconductor is approximated as a linear conductor. In previous work, this model has reproduced the phase transition from homogeneous stationary to homogeneous oscillating states semiquantitatively. In the present work, the formation of spatial patterns is investigated through linear stability analysis and through numerical simulations of the initial value problem; the methods agree well. They show the onset of spatiotemporal patterns for high semiconductor resistance. The parameter dependence of temporal or spatiotemporal pattern formation is discussed in detail.