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The best approach to compute the long-range stray field by micromagnetic simulations of systems with periodic boundary conditions (PBCs) on regular grids is the fast Fourier transform (FFT)-based solution of the Poisson equation combined with the Ewald method to ensure a rapid convergence of the Fourier series. Here, we present the version of such an FFT-Ewald method suitable for grids of rectangular cells. Further, we have incorporated the evaluation of the near-field part of the Ewald sums into the FFT procedure used to evaluate the field of the Gaussian dipole lattice, so that no additional time is spent for the near-field computation. The method described can be used for simulation of any three- or two-dimensional systems with PBC. We present physical examples dealing with extended thin films and arrays of nanowires and nanodots.