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 Nyström 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 Nyström 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 that are typically encountered in the description of one-dimensional closed curves.

中文翻译：

---

亥姆霍兹分解边界积分方程公式的 Nyström 离散的稳定性估计，用于使用狄利克雷边界条件在二维中求解 Navier 散射问题


弹性场的亥姆霍兹分解是解决 Navier 散射问题的常用方法。在边界积分方程 （BIE） 的上下文中使用，该方法通过更简单的亥姆霍兹边界积分算子 （BIO） 提供 Navier 问题的解。亥姆霍兹狄利克雷对诺依曼 （DtN） 的近似值可以在正则化组合场策略中采用，以提供第二种 BIE 公式，用于使用狄利克雷边界条件在二维中求解纳维散射问题，至少在平滑边界的情况下是这样。与散射和传输亥姆霍兹问题的情况不同，我们在亥姆霍兹分解中使用的 DtN 映射的近似值 BIE 在 Navier 情况下需要在其伪微分渐近展开中加入低阶项。这些低阶项存在于 Navier 正则化 BIE 公式中，这使得在全局三角插值和 Kussmaul-Martensen 核奇点分裂策略框架中对其 Nyström 离散的稳定性分析变得复杂。主要困难源于相反阶的伪微分算子的组合，必须通过超出主符号的伪微分展开来小心地执行其 Nyström 离散化。在弧长边界参数化的情况下，误差分析要简单得多，而在一维闭合曲线描述中通常遇到的一般平滑参数化的情况下，误差分析要复杂得多。