Well-posedness and finite element analysis for the elastic scattering problem with a modified DtN map

As one of the most popular artificial boundary conditions, the Dirichlet-to-Neumann (DtN) boundary condition has been widely developed and investigated for solving the exterior wave scattering problems. This work studies the application of a Fourier series DtN map for the elastic scattering problem. The infinite series of the DtN map requires to be truncated in the practical numerical application, and then the well-posedness of the resulting boundary value problem (BVP) becomes a challenging issue. By introducing a corresponding eigensystem to the bilinear form together with appropriate truncated norm estimates, we prove the well-posedness of the corresponding BVP in a weak sense. In addition, a priori error estimates that incorporate the effects of the finite element discretization and the truncation of infinite series are derived. Finally, numerical tests are implemented to validate the theoretical results.