Upper and lower bounds for local electromagnetic quantities

Most engineering problems are solved by means of numerical methods that are able to provide only approximate solutions, for which it would be extremely useful to have e¦cient error estimators. Upper and lower bounds for quantities of integral character, like the stored magnetic energy or the ohmic power dissipated in the domain of interest, had been clearly established along with the procedures to obtain them numerically. However, upper and lower bounds for local quantities would be of paramount interest in several Þelds of applications like Non-Destructive Testing or Nuclear Magnetic Resonance. We present here a procedure for the determination of upper and lower bounds of local Þeld quantities, namely the average value of a Þeld component in an arbitrarily small region. It is based on the introduction of an auxiliary Þeld, and is the natural extension of the method establishing the bounds of global quantities. Our technique can be used for any linear system in stationary conditions for which a virtual work principle can be applied. Its e¦ciency is demonstrated with the analysis of some stationary 2D and 3D electromagnetic problems