DOMAIN DECOMPOSITION PRECONDITIONING FOR HIGH-FREQUENCY HELMHOLTZ PROBLEMS WITH ABSORPTION

作者：Graham I G; Spence E A; Vainikko E

来源：Mathematics of Computation, 2017, 86(307): 2089-2127.

DOI：10.1090/mcom/3190

• Helmholtz equation
• high frequency
• absorption
• iterative solvers
• preconditioning
• domain decomposition
• GMRES

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) + i epsilon)u = f, with absorption parameter epsilon is an element of R. Multigrid approximations of this equation with epsilon not equal 0 are commonly used as preconditioners for the pure Helmholtz case (epsilon = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (epsilon not equal 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 epsilon-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |epsilon| similar to 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 epsilon - 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 O(n(4/3)) for solving finite-element systems of size n - O(k(3)), where we have chosen the mesh diameter h similar to k(-3/2) to avoid the pollution effect. Experiments on problems with h similar to k(-1), i.e., a fixed number of grid points per wavelength, are also given.

• 出版日期2017-9