A remark on the single scattering preconditioner applied to boundary integral equations

TL;DR

This paper analyzes the effects of the single scattering preconditioner on boundary integral equations in multiple scattering problems, showing it simplifies equations and equalizes convergence rates of solvers.

Contribution

It proves that the single scattering preconditioner makes different integral equations identical or equivalent, ensuring uniform convergence behavior.

Findings

Preconditioning makes all direct integral equations identical.

The indirect Brakhage-Werner equation becomes equivalent to direct equations.

Convergence rates of Krylov solvers are the same for all preconditioned equations.

Abstract

This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage-Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper.