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A computational subcell model for thin wires is developed for electromagnetic simulations. The Maxwell equations are discretized by a finite element approximation on a tetrahedral grid. The wires are described by a second-order equation for the current. The geometry of the wires can be chosen independent of the volume grid. A symmetric coupling between field and wires yields a stable semi-discrete field-wire system and an unconditionally stable fully discrete field-wire system. The system of equations is in each time step solved by a preconditioned conjugate gradient method. The accuracy of the subcell model is demonstrated for dipole and loop antennas with comparisons with the method of moments and experimental data.