# New preconditioners for Laplace and Helmholtz integral equations on open curves: Analytical framework and Numerical results

## TL;DR

This paper introduces two novel preconditioners for first-kind integral equations in 2D Helmholtz scattering problems, improving the conditioning of discretized systems through analytical and numerical analysis.

## Contribution

The paper develops and analyzes new preconditioners based on square-roots of local operators for open curves, extending existing analytical preconditioners to Lipschitz scatterers.

## Key findings

- Preconditioners significantly improve system conditioning.
- Numerical examples demonstrate enhanced computational efficiency.
- The analytical framework supports the design of effective preconditioners.

## Abstract

The Helmholtz wave scattering problem by screens in 2D can be recast into first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners, in the form of square-roots of local operators respectively for the corresponding problems with Dirichlet and Neumann conditions on the arc. They generalize the so-called "analytical" preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples.