A new local method for finite difference/finite element (FD/FE) mesh truncation in the frequency domain is investigated. The method is based on the measured equation of invariance (MEI) concept recently proposed for the numerical solution of electromagnetic wave scattering by perfectly conducting targets in unbounded regions. An MEI is a numerically derived discrete, linear equation which relates the field at a given boundary node to the field values at neighboring nodes. The extension of the MEI-based mesh truncation method to treat the case of penetrable scatterers is discussed. A new approach that employs distributions of multipoles to expedite the generation of the MEIs is proposed. This new approach is compared to an earlier one based on the direct application of Huygen's principle. Numerical results are presented for time harmonic scattering by various two-dimensional targets. Since the derivation of the MEI is not based on any far-field assumptions, the mesh truncation condition can be applied just a few cells away from the scatterer's boundary, and a computationally efficient and accurate FD/FE grid truncation can be achieved.