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Abstract
There has been much recent research on preconditioning discretisations of the Helmholtz operator Δ+k2 (subject to suitable boundary conditions) using a discrete version of the socalled “shifted Laplacian” Δ+(k2+iε) for some ε>0 . This is motivated by the fact that, as ε increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large ε can be so that the shifted problem provides a preconditioner that leads to k independent convergence of GMRES, and our main result is a sufficient condition on ε for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the soundsoft scattering problem (with the radiation condition in the latter problem imposed as a farfield impedance boundary condition). Note that we do not address the important question of how large ε should be so that the preconditioner can easily be inverted by standard iterative methods.
Original language  English 

Pages (fromto)  567614 
Number of pages  48 
Journal  Numerische Mathematik 
Volume  131 
Issue number  3 
Early online date  25 Jan 2015 
DOIs  
Publication status  Published  1 Nov 2015 
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Dive into the research topics of 'Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumberindependent convergence is guaranteed?'. Together they form a unique fingerprint.Projects
 2 Finished

Post Doc Fellowship  New Methods and Analysis for Wave Propagation Problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/04/11 → 31/03/14
Project: Research council

Boundary Integral Equation Methods for HF Scattering Problems
Graham, I. (PI) & Smyshlyaev, V. P. (CoI)
Engineering and Physical Sciences Research Council
24/03/09 → 23/09/12
Project: Research council
Profiles

Euan Spence
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching