  We present a method for performing time domain simulations of a microphotonic
system containing a four level gain medium based on the finite element method.
This method includes an approximation that involves expanding the pump and
probe electromagnetic fields around their respective carrier frequencies,
providing a dramatic speedup of the time evolution. Finally, we present a two
dimensional example of this model, simulating a cylindrical spaser array
consisting of a four level gain medium inside of a metal shell.
