The measured equation of invariance (MEI) is a simple technique used to derive finite difference type local equations at mesh boundaries, where the conventional finite difference approach fails (Mei et al., 1992, 1994). Conventionally, finite difference or finite element meshes span from boundary to boundary, or to any surface where an absorbing boundary condition can be simulated. It is demonstrated that the MEI technique can be used to terminate meshes very close to the object boundary and still strictly preserves the sparsity of the finite difference equations. It results in dramatic savings in computing time and memory needs. In an earlier paper (Mei et al., 1992) it was shown that this new method can be applied to general boundary geometries including both convex and concave metal surfaces. In this paper we shall show that the MEI can be applied also to material discontinuities. The method of MEI is based on the postulate that the boundary equations, which govern the scattered fields, are independent of the incident fields. That postulate has been shown to be true for metal scatterers. In a penetrable medium the situation is quite different; in fact, inside the medium the separation of the incident and scattered fields is not a simple matter. Normally, the total field has to be calculated inside the medium, thus violating the postulate. Therefore we cannot use the MEI inside a penetrable medium. The space inside the medium has to be filled by finite difference or finite element meshes. This paper shows to apply the MEI in this situation and concludes that the CPU time required for the MEI method is comparable to that of the method of moments (MOM), when MOM is based on the boundary integral method, i.e., if the scatterer is homogeneous. However, for inhomogeneous penetrable scatterer the MEI method definitely is preferable to the MOM. ¿ American Geophysical Union 1994