Error-based derivation of complementary formulations for the eddy-current problem

An error-based approach is developed for complementary formulations of eddy-current problems corresponding to a mismatch in constitutive relationships. The derivation accounts systematically for the various problem specifications, including boundary conditions. It is shown that under steady harmonic conditions energy bounds cannot be established; the problem gives rise to complementary variational principles that are not extremal, as well as a single extremal principle that involves both complementary fields simultaneously. The treatment results in a general framework that places previously proposed formulations in proper perspective, and that accounts for differences among them.