It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method, we analyze and solve a two dimensional transmission problem describing elastic waves inside a bounded and closed elastic obstacle and acoustic waves outside it. We are mainly interested in analyzing the DtN boundary condition of the Helmholtz equation in order to establish the well-posedness results of the approximated variational equation, and further derive a priori error estimates involving effects of both the finite element discretization and the truncation of DtN map. Finally, some numerical results are presented to illustrate the accuracy of the numerical scheme.

众所周知，傅立叶级数Dirichlet-Neumann（DtN）边界条件可用于求解无界域中的Helmholtz方程。在这项工作中，应用此类DtN边界条件并使用有限元方法，我们分析并解决了二维传输问题，该问题描述了有界和封闭的弹性障碍物内部的弹性波以及其外部的声波。我们主要对Helmholtz方程的DtN边界条件进行分析，以建立近似变分方程的适定性结果，并进一步推导涉及有限元离散化和DtN映射截断的先验误差估计。最后，给出了一些数值结果，说明了数值方案的准确性。