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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 
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
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council