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The efficient computation of large eddy-current problems with finite elements requires adaptive methods and fast optimal iterative solvers such as multigrid methods. This paper provides an overview of the most important implementation aspects of an adaptive multigrid scheme for time-harmonic eddy currents. Numerical experiments show that the standard multigrid scheme can be modified to yield an O(N) complexity even for general adaptive refinement strategies, where the number of unknowns N can grow slowly from one to the next refinement level. Algorithmic details and numerical examples are given.