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. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then 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.

我们认为 GMRES 应用于具有强陷波的高频亥姆霍兹方程的离散化；回想一下，在这种情况下，随着频率序列的增加，问题呈指数病态。我们的主要关注点是 2 维和 3 维外部 Dirichlet 和 Neumann 障碍问题的边界积分方程公式。在关于积分算子特征值分布的某些假设下，我们证明了 GMRES 迭代次数如何随频率增长的上限；然后我们在数值上研究我们的边界和边界的锐度（根据频率的依赖性）各种数量进入我们的范围。因此，本文首次全面研究了俘获下亥姆霍兹边界积分方程的 GMRES 迭代次数的频率依赖性。