Dielectric breakdown during high power operation is hazardous to electric and electronic devices and systems. During the breakdown process, the bound charges break free and are pushed to move under the force of high intensity fields. As a result, the dielectric which was a good insulator becomes a conductor. Such a process can be described as the change of conductivity of the dielectric, which in this case, is a nonlinear function of the electric field. In this paper, the nonlinear conductivity is incorporated into Maxwell's equations, and the resulting nonlinear equation is solved using the time-domain finite element method together with Newton's method. A numerical example is presented to demonstrate the capability of the proposed algorithm and the nonlinear effect caused by the nonlinear conductivity.