We consider a nonlinear elliptic-parabolic system which stems from the modelling of heat induction processes in which a sinusoidal electromagnetic field is applied at the boundary of the inductor. The magnetic field satisfies an elliptic equation whereas the temperature equation is parabolic. These two equations are coupled due to the Joule effect and to the conductivity dependence on temperature. A Galerkin method for our problem is analyzed and shown to yield error estimates in L-2 and H-1.