  Waves scattering from unbounded structures are always complicated problems
for numerical simulations. For the case that the non-periodic incident field
scattered by (locally perturbed) periodic surfaces, with the help of the Bloch
transform, the problem could be solved by some finite element methods, if the
incident fields decay at certain rate at the infinity. For faster decaying
incident fields, a high order numerical method is also available. However, in
these cases, the plain waves, which belong to a very important family of
incident fields but do not decay at the infinity, are not included. In this
paper, we aim to develop the Bloch transform based standard finite method for
this certain case, and then establish the high order method afterwards.
Numerical experiments have been carried out for both the standard and high
order numerical methods. Based on the algorithms for incident plain waves, we
could also extend the numerical methods to more generalized cases when only the
not so efficient standard method is available.
