In this paper, the homotopy deformation method to solve the nonlinear stationary semiconductor equations with Fermi-Dirac statistic is used. This method introduces an artificial transient problem. The time discretization is based on the nonlinear implicit scheme with local time steps. In order to have an automatic adaptation of local time step parameters, we introduce arclength predictor-c- orrector continuation methods. The fondamental goal of these methods is to overcome the unstabilities or the failure of the classical Newton-Raphson's schemes which appear when the nonlinearity is Strong or near Limit or Bifurcation points. The approximate procedure of our system using a Galerkin method that makes use of a mixed finite element approach is used. A peculiar feature of this mixed formulation is that the electric displacement $D$ and the current densities $j_n$ and $j_p$ for electrons and holes, are taken as unknowns, together with the potential $\phi$ and quasi_Fermi levels $\phi_n$ and $\phi_p$. This allows $D$, $j_n$ and $j_p$ to be determined directly and accurately. The above algorithms appear to be efficient, robust and to give satisfactory results. Numerical results are presented, in one and two dimension, for some realistic devices : an Heterojunction Diode (quasi-1D problem) and an Heterojunction Bipolar Transistor (HBT) working in amplifier mode.