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Abstract
A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate hversion Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as k→∞ h must decrease with k at the same rate as for the standard formulation). We prove kexplicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positivedefinite matrix. Even though the number of iterations grows with k, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positivedefinite matrix.
Original language  English 

Pages (fromto)  110131 
Number of pages  22 
Journal  Journal of Computational and Applied Mathematics 
Volume  352 
Early online date  6 Dec 2018 
DOIs  
Publication status  Published  15 May 2019 
Keywords
 math.NA
 35J05, 65N30, 65F10
 Finite element method
 GMRES
 Helmholtz equation
 Wavenumberexplicit analysis
 Coercive variational formulation
 Pollution effect
ASJC Scopus subject areas
 Computational Mathematics
 Applied Mathematics
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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