Validating Gauss's law in inhomogeneous medium

I have a set of finite-difference time-domain (FDTD) simulated time harmonic electric field (E) and current density (J). The data domain is in an inhomogeneous dielectric medium. Based on Gauss Law and charge conservation, the following should be valid: $iomeganablacdot{E}=nablacdot{J}$. However, in my data the result is far from being accurate. Is this because of the simulation’s precision, which I have no idea how the software restrains its algorithm based on the Gauss Law? Or did I miss some fundamental physics?
Thanks!