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Abstract
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist nearzero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior DirichlettoNeumann map on the truncation boundary). Our motivation for proving this result is that (a) the finiteelement method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and (b) the location of eigenvalues, and especially nearzero ones, plays a key role in understanding how iterative solvers such as the generalized minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finiteelement method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretizations of highfrequency Helmholtz problems under strong trapping (the subject of the companion paper [P. Marchand et al., Adv. Comput. Math., to appear]).
Original language  English 

Journal  SIAM Journal on Mathematical Analysis 
Volume  53 
Issue number  6 
Early online date  30 Nov 2021 
DOIs  
Publication status  Published  31 Dec 2021 
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Dive into the research topics of 'Eigenvalues of the truncated Helmholtz solution operator under strong trapping'. Together they form a unique fingerprint.Projects
 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