A finite difference approximation to the semiconductor device equations using the Bernoulli function approximation to the exponential function is described, and the robustness of this approximation is demonstrated. Sheikh's convergence analysis of Gummel's method and quasi-newton methods is extended to a nonunform mesh and the Bernoulli function discretization. It is proved that Gummel's method and the quasi-newton methods for scaled carrier densities and carrier densities converge locally for sufficiently smooth problems. A new numerical method for semiconductor device simulation is also presented. The device is covered witha relatively coarse nonuniform grid, and Newton's method is applied globally over the entire grid and locally over critical retangular grid subsets where the resdual is large until a solution of moderate accuracy is obtained. Rectangular subgrids of the critical subgrids are then refined by mesh-halving into coextensive sets of grids, and an accurate solution isobtained on the finest subgrids by applying the multigrid method directly to the corresponding sets of nonlinear difference equations. The method is fast because the Newton iteration is applied mainly to small critical regions of the device and the multigrid method significantly reduces the time required to obtain an accurate solution on the finest grids in these regions. The method is applied to a short-channel MOSFET and is hown to copare favorably with conventional methos