<p>The computation of the propagation of an electromagnetic wave in the time domain is examined using local Faber polynomial based time dependent propagators. Conventionally, the whole computational domain is evaluated by one global operator. Contrary, when utilizing a nonuniform discretization in the system local operators can be used individually for each subarea. This allows the complexity to be reduced by decreasing the polynomial order of the evaluation of the Faber algorithm, while at the same time decreasing the overall runtime. Compared to common Local Time Step methods, the time step size of each individual area with this approach is already synchronized with a predefined global time step size. In general, the investigated approach is especially interesting for applications that demand a high spatial resolution, such as in the field of nanophotonics and THz-technology. However, the influence of the necessary process steps on the runtime must be examined in particular when computing with the local operators approach. To this end, the theoretical complexity is derived and compared with practical results to analyze the efficiency.</p>