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

Document Type

Article

Publication Date

12-1-2024

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 et al. (2021) [20]). 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 et al. (2015) [6] 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 Nyström discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers.

Identifier

85204775110 (Scopus)

Publication Title

Computers and Mathematics with Applications

External Full Text Location

https://doi.org/10.1016/j.camwa.2024.09.013

ISSN

08981221

First Page

152

Last Page

173

Volume

175

Grant

DMS-1908602

Fund Ref

Ministerio de Ciencia e Innovación

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