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It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundarycondition can be used to solve the Helmholtz equation in unbounded domains. Inthis work, applying such DtN boundary condition and using the finite elementmethod, we solve and analyze a two dimensional transmission problem describingelastic waves inside a bounded and closed elastic obstacle and acoustic wavesoutside it. We are mainly interested in analyzing the DtN boundary condition ofthe Helmholtz equation in order to establish the well-posedness results of theapproximated variational equation, and further derive a priori error estimatesinvolving effects of both the finite element discretization and the DtNboundary condition truncation. Finally, some numerical results are presented toillustrate the accuracy of the numerical scheme.

Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods