  Consider the scattering of a time-harmonic elastic plane wave by a periodic
rigid surface. The elastic wave propagation is governed by the two-dimensional
Navier equation. Based on a Dirichlet-to-Neumann (DtN) map, a transparent
boundary condition (TBC) is introduced to reduce the scattering problem into a
boundary value problem in a bounded domain. By using the finite element method,
the discrete problem is considered, where the TBC is replaced by the truncated
DtN map. A new duality argument is developed to derive the a posteriori error
estimate, which contains both the finite element approximation error and the
DtN truncation error. An a posteriori error estimate based adaptive finite
element algorithm is developed to solve the elastic surface scattering problem.
Numerical experiments are presented to demonstrate the effectiveness of the
proposed method.
