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Abstract
We consider GMRES applied to discretisations of the highfrequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially illconditioned through an increasing sequence of frequencies. Our main focus is on boundaryintegralequation formulations of the exterior Dirichlet and Neumann obstacle problems in 2 and 3d. 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 frequencydependence of the number of GMRES iterations for Helmholtz boundaryintegral equations under trapping.
Original language  English 

Article number  37 
Number of pages  63 
Journal  Advances in Computational Mathematics 
Volume  48 
Issue number  4 
Early online date  4 Jun 2022 
DOIs  
Publication status  Published  31 Aug 2022 
Bibliographical 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 PierreHenri Tournier (Sorbonne Université, 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 PierreHenri Tournier (Sorbonne Université, 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:
© 2022, The Author(s).
Keywords
 GMRES
 Helmholtz equation
 High frequency
 Trapping
ASJC Scopus subject areas
 Computational Mathematics
 Applied Mathematics
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 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council