Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?

A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate h-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly t...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2019-05, Vol.352, p.110-131
Main Authors: Diwan, Ganesh C., Moiola, Andrea, Spence, Euan A.
Format: Article
Language:English
Subjects:
Coercive variational formulation
Finite element method
GMRES
Helmholtz equation
Pollution effect
Wavenumber-explicit analysis
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Summary:A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate h-version 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 k-explicit 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 positive-definite 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 positive-definite matrix.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2018.11.035