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Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption


Authors: I. G. Graham, E. A. Spence and E. Vainikko
Journal: Math. Comp. 86 (2017), 2089-2127
MSC (2010): Primary 35J05, 65N55; Secondary 65F08, 65F10, 65N30, 78A45
DOI: https://doi.org/10.1090/mcom/3190
Published electronically: February 8, 2017
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Abstract: In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise linear finite-element approximations of the Helmholtz equation $ -\Delta u - (k^2+ \textup {i} \varepsilon )u = f$, with absorption parameter $ \varepsilon \in \mathbb{R}$. Multigrid approximations of this equation with $ \varepsilon \not = 0$ are commonly used as preconditioners for the pure Helmholtz case ( $ \varepsilon = 0$). However a rigorous theory for such (so-called ``shifted Laplace'') preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ( $ \varepsilon \not =0$), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a $ k$- and $ \varepsilon $-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $ \vert\varepsilon \vert\sim k^2$, then classical overlapping Additive Schwarz will perform optimally for the damped problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case $ \varepsilon = 0$. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about $ \mathcal {O}(n^{4/3})$ for solving finite-element systems of size $ n=\mathcal {O}(k^3)$, where we have chosen the mesh diameter $ h \sim k^{-3/2}$ to avoid the pollution effect. Experiments on problems with $ h\sim k^{-1}$, i.e., a fixed number of grid points per wavelength, are also given.


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Additional Information

I. G. Graham
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: I.G.Graham@bath.ac.uk

E. A. Spence
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: E.A.Spence@bath.ac.uk

E. Vainikko
Affiliation: Institute of Computer Science, University of Tartu, 50409, Estonia
Email: eero.vainikko@ut.ee

DOI: https://doi.org/10.1090/mcom/3190
Keywords: Helmholtz equation, high frequency, absorption, iterative solvers, preconditioning, domain decomposition, GMRES
Received by editor(s): July 7, 2015
Received by editor(s) in revised form: March 23, 2016
Published electronically: February 8, 2017
Article copyright: © Copyright 2017 American Mathematical Society