There is a large body of literature investigating the static and dynamic properties of polyelectrolytes due both to their widespread application in industrial processes and their ubiquitous presence in biology. Because of their highly charged nature, polyelectrolytes tend to alter the local dielectric permittivity of the solution within a few nanometers of their backbone. This effect has, however, been almost entirely ignored in both simulations and theoretical work. In this article, we apply our recently developed electrostatic solver based on Maxwell's equations to examine the effects of the permittivity reduction in the vicinity of the polyelectrolyte. We first verify our new approach by calculating and comparing ion distributions around a linear fixed polyelectrolyte and find both quantitative and qualitative changes in the ion distribution. Further simulations with an applied electric field show that the reduction in the local dielectric constant increases the mobility of the chains by approximately ten percent. More importantly, variations in the local dielectric constant lead to qualitatively different behavior of the conductivity.