Robust boundary integral equations for the solution of elastic scattering problems via Helmholtz decompositions

TL;DR

This paper develops and analyzes new second-kind boundary integral equations for elastic scattering problems using Helmholtz decompositions, improving robustness and computational efficiency for smooth scatterers in 2D.

Contribution

It introduces a regularization methodology to create second-kind BIE formulations for elastic scattering via Helmholtz decompositions, enhancing stability and solver performance.

Findings

Second-kind BIE formulations show improved robustness.

Numerical results demonstrate efficient iterative solver performance.

Method applicable to smooth scatterers in two dimensions.

Abstract

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.