We first develop efficient and robust adaptive edge element
methods for the Maxwell cavity problem in 3D which are of optimal
computational complexity. Then we report our recent efforts in developing
adaptive PML method for solving time-harmonic scattering problems.
The adaptive PML method provides a complete way to solve
the scattering problem with error control in the FE framework in
which the total computational costs are insensitive to the thickness
of the PML absorbing layer. This talk is based on the joint work
with Tao Cui and Linbo Zhang.