A Non-Overlapping Domain Decomposition Method for the Exterior Helmholtz Problem
نویسندگان
چکیده
In this paper, we first show that the domain decomposition methods that are usually efficient for solving elliptic problems typically fail when applied to acoustics problems. Next, we present an alternative domain decomposition algorithm that is better suited for the exterior Helmholtz problem. We describe it in a formalism that can use either one or two Lagrange multiplier fields for solving the corresponding interface problem by a Krylov method. In order to improve convergence and ensure scalability with respect the number of subdomains, we propose two complementary preconditioning techniques. The first preconditioner is based on a spectral analysis of the resulting interface operator and targets the high frequency components of the error. The second preconditioner is based on a coarsening technique, employs plane waves, and addresses the low frequency components of the error. Finally, we show numerically that, using both preconditioners, the convergence rate of the proposed domain decomposition method is quasi independent of the number of elements in the mesh, the number of subdomains, and depends only weakly on the wavenumber, which makes this method uniquely suitable for solving large-scale high frequency exterior acoustics problems. Acoustic wave propagation problems lead to linear systems that become very large in the high frequency regime. Indeed, for most discretization methods, the mesh size h is typically chosen as one tenth of the wavelength in order to ensure a basic approximation of the physical phenomena. For this reason, many iterative solvers have been and continue to be developed for the Helmholtz problem. In this paper, we consider a domain decomposition based iterative algorithm, because of the success encountered by such methods for the solution of elliptic problems, and because they can be easily implemented on parallel computers.
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تاریخ انتشار 1998