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.

中文翻译：

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使用改进的 DtN 映射进行弹性散射问题的适定性和有限元分析


作为最流行的人工边界条件之一，狄利克雷-诺依曼 （DtN） 边界条件已被广泛开发和研究，用于求解外波散射问题。这项工作研究了傅里叶级数 DtN 映射在弹性散射问题中的应用。在实际数值应用中，需要截断 DtN 映射的无限级数，然后所得边界值问题 （BVP） 的适定性成为一个具有挑战性的问题。通过将相应的特征系统引入双线性形式以及适当的截断范数估计，我们在弱意义上证明了相应 BVP 的适定性。此外，还推导出了包含有限元离散化和无限级数截断的影响的先验误差估计。最后，通过数值测试验证了理论结果。