Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption
نویسندگان
چکیده
In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation −∆u − (k2 + iε)u = f , with absorption parameter ε ∈ R. Multigrid approximations of this equation with ε 6= 0 are commonly used as preconditioners for the pure Helmholtz case (ε = 0). However a rigorous theory for such (so-called “shifted Laplace”) preconditioners, either for the pure Helmholtz equation or even the damped equation, is still missing. We present a new theory for the damped equation that provides rates of convergence for (leftor right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a kand ε-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |ε| ∼ k2, then classical overlapping additive Schwarz will perform optimally for the damped problem, provided the coarse mesh diameter is carefully chosen. Extensive numerical experiments illustrate the theory and also give insight into how domain decomposition approximations of the damped problem perform as preconditioners for the pure Helmholtz case. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical complexity of about O(n4/3) for solving finite element systems of size n = O(k3), where we have chosen the mesh diameter h ∼ k−3/2 to avoid the pollution effect. Experiments on problems with h ∼ k−1, i.e. a fixed number of grid points per wavelength, are also given.
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ورودعنوان ژورنال:
- Math. Comput.
دوره 86 شماره
صفحات -
تاریخ انتشار 2017