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We have adapted our finite element micromagnetic simulation software to the massively parallel architecture of graphical processing units (GPUs) with double-precision floating point accuracy. Using the example of Standard Problem #4 with different numbers of discretization points, we demonstrate the high speed performance of a single GPU compared with an OpenMP-parallelized version of the code using eight CPUs. The adaption of both the magnetostatic field calculation and the time integration of the Landau-Lifshitz-Gilbert equation routines can lead to a speedup factor of up to four. The gain in computation performance of the GPU code increases with increasing number of discretization nodes. The computation time required for high-resolution micromagnetic simulations of the magnetization dynamics in large magnetic samples can thus be reduced effectively by employing GPUs.