On the conditioning of the steady state semiconductor device problem

In this paper we carry out a conditioning analysis for the steady state semiconductor device problem. We consider various quasilinearizations as well as Gummel-type iterations and obtain stability bounds which may allow ill-conditioning' in general. These bounds are exponential in the potential variation, and are sharp e.g. for a thyristor. But for devices where each smooth subdomain has an Ohmic contact, e.g. a pn-diode, moderate bounds guaranteeing well-conditioning are obtained. Moreover, the analysis suggests how various row and column scalings should be applied in order for the measured condition numbers of the linearized discrete problem to correspond more realistically to the true loss of significant digits in the calculations.