Helmholtz decompositions of the elastic fields open up new avenues for the solution of linear elastic scattering problems via boundary integral equations (BIE) [Dong, Lai, Li, Mathematics of Computation,2021]. The main appeal of this approach is that the ensuing systems of BIE feature only integral operators associated with the Helmholtz equation. However, these BIE involve non standard boundary integral operators that do not result after the application of either the Dirichlet or the Neumann trace to Helmholtz single and double layer potentials. Rather, the Helmholtz decomposition approach leads to BIE formulations of elastic scattering problems with Neumann boundary conditions that involve boundary traces of the Hessians of Helmholtz layer potential. As a consequence, the classical combined field approach applied in the framework of the Helmholtz decompositions leads to BIE formulations which, although robust, are not of the second kind. Following the regularizing methodology introduced in [Boubendir, Dominguez, Levadoux, Turc, SIAM Journal on Applied Mathematics 2015] we design and analyze novel robust Helmholtz decomposition BIE for the solution of elastic scattering that are of the second kind in the case of smooth scatterers in two dimensions. We present a variety of numerical results based on Nystrom discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers.

翻译：弹性场的亥姆霍兹分解为通过边界积分方程（BIE）求解线性弹性散射问题开辟了新途径[Dong, Lai, Li, Mathematics of Computation, 2021]。该方法的主要优势在于其衍生的BIE系统仅包含与亥姆霍兹方程相关的积分算子。然而，这些BIE涉及非标准的边界积分算子，它们并非通过对亥姆霍兹单层与双层势应用狄利克雷迹或诺伊曼迹直接得到。相反，亥姆霍兹分解方法导出的诺伊曼边界条件下弹性散射问题的BIE表述，涉及亥姆霍兹层势海森矩阵的边界迹。因此，在亥姆霍兹分解框架下应用经典组合场方法所产生的BIE表述虽然具有鲁棒性，却并非第二类方程。遵循[Boubendir, Dominguez, Levadoux, Turc, SIAM Journal on Applied Mathematics 2015]中引入的正则化方法，我们针对二维光滑散射体情形，设计并分析了一类新型的、具有鲁棒性的第二类亥姆霍兹分解BIE，用于求解弹性散射问题。基于Nyström离散化的多种数值结果表明，第二类正则化公式在与迭代求解器结合时表现出良好的性能。