Second-Kind integral solvers for TE and TM problems of diffraction by open-arcs

TL;DR

This paper introduces a new numerical method for solving diffraction problems involving open arcs in 2D, achieving high accuracy efficiently for both TE and TM polarizations.

Contribution

It develops second-kind integral equations for open-arc diffraction problems using weighted operators and a generalized Calderón formula, enhancing computational efficiency.

Findings

High accuracy results with few iterations

Effective for both low and high frequencies

Reduced computational time

Abstract

We present a novel approach for the numerical solution of problems of diffraction by open arcs in two dimensional space. Our methodology relies on composition of {\em weighted versions} of the classical integral operators associated with the Dirichlet and Neumann problems (TE and TM polarizations, respectively) together with a generalization to the open-arc case of the well known closed-surface Calder\'on formulae. When used in conjunction with spectrally accurate discretization rules and Krylov-subspace linear algebra solvers such as GMRES, the new second-kind TE and TM formulations for open arcs produce results of high accuracy in small numbers of iterations and short computing times---for low and high frequencies alike.