Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

TL;DR

This paper investigates how the number of GMRES iterations needed to solve high-frequency Helmholtz equations with trapping depends on frequency, providing theoretical bounds and numerical validation for boundary-integral formulations.

Contribution

It offers the first comprehensive analysis of GMRES iteration growth with frequency for Helmholtz boundary-integral equations under trapping conditions.

Findings

Derived upper bounds on GMRES iterations growth with frequency.

Numerical experiments confirm the sharpness of theoretical bounds.

First study to analyze frequency dependence of GMRES for trapped Helmholtz problems.

Abstract

We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.