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Consider the scattering of a time-harmonic elastic plane wave by abi-periodic rigid surface. The displacement of elastic wave motion is modeledby 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 infiniteseries, an exact transparent boundary condition is introduced and thescattering problem is formulated equivalently into a boundary value problem ina bounded domain. An a posteriori error estimate based adaptive finite elementDtN method is proposed to solve the discrete variational problem where the DtNoperator is truncated into a finite number of terms. The a posteriori errorestimate takes account of the finite element approximation error and thetruncation error of the DtN operator which is shown to decay exponentially withrespect to the truncation parameter. Numerical experiments are presented toillustrate the effectiveness of the proposed method.

An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures