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The standard adaptive edge finite element method (AEFEM), using first/second family Nédélec edge elements with any order, for the three‐dimensional H(curl)‐elliptic problems with variable coefficients is shown to be convergent for the sum of the energy error and the scaled error estimator. The special treatment of the data oscillation and the interior node property are removed from the proof. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi‐optimal. Copyright © 2010 John Wiley & Sons, Ltd.