Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of Boundary Integral Equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nystr\"om discretizations in the framework of global trigonometric interpolation and the Kussmaul-Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nystr\"om discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations which are typically encountered in the description of one dimensional closed curves.

翻译：弹性场的Helmholtz分解是求解Navier散射问题的常用方法。在边界积分方程框架下，该方法通过更简单的Helmholtz边界积分算子实现Navier问题的求解。结合正则化组合场策略，可采用Helmholtz Dirichlet-to-Neumann（DtN）映射的近似形式，为具有Dirichlet边界条件的二维Navier散射问题（至少针对光滑边界情形）提供第二类边界积分方程公式。与Helmholtz散射及透射问题不同，我们在Navier情形下Helmholtz分解边界积分方程中使用的DtN映射近似，需要在拟微分渐近展开中纳入低阶项。这些低阶项的存在使Navier正则化边界积分方程公式的Nyström离散化稳定性分析变得复杂——该离散化基于全局三角插值框架与Kussmaul-Martensen核奇异性分裂策略。主要困难源于相反阶拟微分算子复合的Nyström离散化处理，需通过主象征之外的拟微分展开谨慎实现。误差分析在弧长边界参数化情形下显著简化，而在描述一维闭合曲线时常见的一般光滑参数化情形下则复杂得多。