@article{628a7b4b949b498f964249e5f81bbc19,

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

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. 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.",

keywords = "GMRES, Helmholtz equation, High frequency, Trapping",

author = "Pierre Marchand and Jeffrey Galkowski and Alastair Spence and Spence, {Euan A.}",

note = "Funding Information: EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne Universit{\'e}, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1. Funding Information: EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne Universit{\'e}, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

year = "2022",

month = aug,

day = "31",

doi = "10.1007/s10444-022-09931-9",

language = "English",

volume = "48",

journal = "Advances in Computational Mathematics",

issn = "1019-7168",

publisher = "Springer Netherlands",

number = "4",

}