. The convergence of the original iteration is governed by the spectral radius [T

We compare both numerically and theoretically three techniques for accelerating the conver-gence of a nonlinear xed point iteration arising from a system of coupled partial dierential equa-tions: Chebyshev acceleration, a second order stationary method, and a nonlinear version of the Generalized Minimal Residual Algorithm (GMRES) which we call NLGMR. All three approaches are implemented in `Jacobian-free ' mode, i.e., only a subroutine which returns T (u) as a function of u is required. We present a set of numerical comparisons for the drift-diusion semiconductor model. For the mapping T which corresponds to the nonlinear block Gau-Seidel algorithm for the solution of this nonlinear elliptic system, NLGMR is found to be superior to the second order stationary method and the Chebychev acceleration. We analyze the local convergence of the non-linear iterations in terms of the spectrum [T u (u)] of the derivative