# Weighted integral solvers for elastic scattering by open arcs in two dimensions

## TL;DR

This paper introduces a new numerical method for elastic scattering problems involving open arcs in two dimensions, combining weighted operators, a novel Calderón relation, and spectral quadrature for high accuracy and efficiency.

## Contribution

It develops a novel approach using weighted operators and a new Calderón relation for elastic scattering by open arcs, with demonstrated high accuracy and efficiency.

## Key findings

- High-accuracy results with few iterations
- Effective for both low and high frequencies
- Numerical examples confirm efficiency

## Abstract

We present a novel approach for the numerical solution of problems of elastic scattering by open arcs in two dimensions. Our methodology relies on the composition of weighted versions of the classical operators associated with Dirichlet and Neumann boundary conditions in conjunction with a certain "open-arc elastic Calder\'on relation" whose validity is demonstrated in this paper on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. Using this Calder\'on relation in conjunction with spectrally accurate quadrature rules and the Krylov-subspace linear algebra solver GMRES, the proposed overall open-arc elastic solver produces results of high accuracy in small number of iterations---for low and high frequencies alike. A variety of numerical examples in this paper demonstrate the accuracy and efficiency of the proposed methodology.