  Consider the scattering of a time-harmonic elastic plane wave by a
bi-periodic rigid surface. The displacement of elastic wave motion is modeled
by the three-dimensional Navier equation in an open domain above the surface.
Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite
series, an exact transparent boundary condition is introduced and the
scattering problem is formulated equivalently into a boundary value problem in
a bounded domain. An a posteriori error estimate based adaptive finite element
DtN method is proposed to solve the discrete variational problem where the DtN
operator is truncated into a finite number of terms. The a posteriori error
estimate takes account of the finite element approximation error and the
truncation error of the DtN operator which is shown to decay exponentially with
respect to the truncation parameter. Numerical experiments are presented to
illustrate the effectiveness of the proposed method.
