  An adaptive finite element method is presented for the elastic scattering of
a time-harmonic plane wave by a periodic surface. First, the unbounded physical
domain is truncated into a bounded computational domain by introducing the
perfectly matched layer (PML) technique. The well-posedness and exponential
convergence of the solution are established for the truncated PML problem by
developing an equivalent transparent boundary condition. Second, an a
posteriori error estimate is deduced for the discrete problem and is used to
determine the finite elements for refinements and to determine the PML
parameters. Numerical experiments are included to demonstrate the competitive
behavior of the proposed adaptive method.
