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：弹性场的亥姆霍兹分解是求解纳维散射问题的常用方法。在边界积分方程（BIE）框架下，该方法通过更简单的亥姆霍兹边界积分算子（BIO）实现纳维问题的求解。在正则化组合场策略中，可利用亥姆霍兹狄利克雷-诺伊曼（DtN）映射的逼近，为二维狄利克雷边界条件下的纳维散射问题（至少对光滑边界情形）建立第二类BIE公式。与亥姆霍兹散射和透射问题不同，纳维情形下亥姆霍兹分解BIE所使用的DtN映射逼近，需在其拟微分渐近展开式中纳入低阶项。这些低阶项在纳维正则化BIE公式中的存在，使得基于全局三角插值和Kussmaul-Martensen核奇异性分裂策略的Nyström离散化稳定性分析复杂化。主要困难源于具有相反阶数的拟微分算子复合，其Nyström离散化必须通过超越主符号的拟微分展开谨慎执行。弧长边界参数化情形下的误差分析显著简化，但通常用于描述一维闭合曲线的通用光滑参数化情形则复杂得多。