Abstract
In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewiselinear finiteelement approximations} of the Helmholtz equation $\Delta u  (k^2+ \ri \eps)u = f$, with absorption parameter $\eps \in \R$. Multigrid approximations of this equation with $\eps \not= 0$ are commonly used as preconditioners for the pure Helmholtz case ($\eps = 0$). However a rigorous theory for such (socalled ``shifted Laplace'') preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ($\eps \not=0$), 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 $\eps$explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $\eps\sim k^2$, 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 $\eps = 0$. 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.
Original language  English 

Pages (fromto)  20892127 
Journal  Mathematics of Computation 
Volume  86 
Issue number  307 
Early online date  8 Feb 2017 
DOIs  
Publication status  Published  2017 
Keywords
 math.NA
 35J05, 65N55, 65F08, 65F10, 65N30, 78A45
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Ivan Graham
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching