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A recursive convolution (RC) based on the vector fitting (VF) method and triangular temporal basis functions is employed to compute the Floquet absorbing boundary condition (FABC) formulation in the vector 3D finite element time-domain (FETD) modelling of doubly periodic structures. This novel implementation (VF-RC-FETD-FABC) results in significantly lower computation time requirements than the standard convolution (SC) implementation. The numerical examples presented show a reduction in computation time requirements by a factor of at least 3.5. In addition, a time window in excess of 50 000 time steps is recorded over which practically stable results are obtained. This temporal window is sufficiently large to allow the modelling of many practical problems. The results from four such problems are presented, confirming the accuracy and speed of the VF-RC-FETD-FABC software.