Spectral Galerkin method for solving elastic wave scattering problems with multiple open arcs

TL;DR

This paper introduces a fast spectral Galerkin method using weighted Chebyshev polynomials for solving elastic wave scattering problems on unbounded domains with multiple open arcs, achieving exponential convergence.

Contribution

The paper develops a novel spectral Galerkin approach with weighted Chebyshev bases for elastic wave scattering on open arcs, ensuring exponential convergence and robustness.

Findings

Method achieves exponential convergence for analytic sources and geometries.

Numerical results confirm high accuracy and robustness across multiple arcs and wavenumbers.

Approach effectively solves weakly- and hyper-singular boundary integral equations.

Abstract

We study the elastic time-harmonic wave scattering problems on unbounded domains with boundaries composed of finite collections of disjoints finite open arcs (or cracks) in two dimensions. Specifically, we present a fast spectral Galerkin method for solving the associated weakly- and hyper-singular boundary integral equations (BIEs) arising from Dirichlet and Neumann boundary conditions, respectively. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions' edge behavior. We show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arcs geometries. Numerical examples demonstrate the accuracy and robustness of the proposed method with respect to number of arcs and wavenumber.