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Abstract
We consider onelevel additive Schwarz preconditioners for the Helmholtz equation (with increasing wavenumber $k$), discretized using fixedorder nodal conforming finite elements on a family of simplicial fine meshes with diameter $h$, chosen to maintain accuracy as $k$ increases. The preconditioners combine independent local solves (with impedance boundary conditions) on overlapping subdomains of diameter $H$ and overlap $\delta$, and prolongation/restriction operators defined using a partition of unity, this formulation was previously proposed in [J.H. Kimn and M. Sarkis, Comp. Meth. Appl. Mech. Engrg. 196, 15071514, 2007]. In numerical experiments (with $\delta \sim H$) we observe robust (i.e. $k$independent) GMRES convergence as $k$ increases, both with $H$ fixed, and with $H$ decreasing moderately as $k$ increases. This provides a highlyparallel, $k$robust onelevel domaindecomposition method. We provide supporting theory for this observation by studying the preconditioner applied to a range of absorptive problems, $k^2\mapsto k^2+ \mathrm{i} \varepsilon$, with absorption parameter $\varepsilon$, including the "pure Helmholtz" case ($\varepsilon = 0$). Working in the Helmholtz "energy" inner product, we prove a robust upper bound on the norm of the preconditioned matrix, valid for all $\varepsilon, \delta$. Under additional conditions on $\varepsilon$ and $\delta$, we also prove a strictlypositive lower bound on the distance of the field of values of the preconditioned matrix from the origin. Using these results, combined with previous results of [M.J. Gander, I.G. Graham and E.A. Spence, Numer. Math. 131(3), 567614, 2015] we obtain theoretical support for the observed robustness of the preconditioner for the pure Helmholtz problem with increasing wavenumber $k$.
Original language  English 

Pages (fromto)  2515–2543 
Journal  SIAM Journal on Numerical Analysis 
Volume  58 
Issue number  5 
Early online date  16 Sep 2020 
DOIs  
Publication status  Published  2020 
Keywords
 math.NA
 65F08, 65F10, 65N55
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Dive into the research topics of 'Domain Decomposition with local impedance conditions for the Helmholtz equation with absorption'. Together they form a unique fingerprint.Projects
 2 Active

Fast solvers for frequencydomain wavescattering problems and applications
Graham, I., Gazzola, S. & Spence, E.
Engineering and Physical Sciences Research Council
1/01/19 → 31/12/22
Project: Research council

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
Profiles

Ivan Graham
 Department of Mathematical Sciences  Professor Emeritus
Person: Honorary / Visiting Staff