Due to the nonlinear structure,traditional approach for error estimates of fully discrete finite element methods for nonlinear parabolic equations often require certain restrictions on the grid ratio(between the time-step size and spatial mesh size),not only for explicit time-stepping schemes but also for implicit or semi-implicit schemes.We introduce a new approach to analyze the discretization errors,which decouples the errors from the temporal and spatial directions and avoid the restrictive conditions on the grid ratio.Here we discuss recent advance of this new approach.