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A non-overlapping domain decomposition method for the exterior Helmholtz problem. (English) Zbl 0909.65103

Mandel, Jan (ed.) et al., Domain decomposition methods 10. The 10th international conference, Boulder, CO, USA, August 10–14, 1997. Providence, RI: AMS, American Mathematical Society. Contemp. Math. 218, 42-66 (1998).
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.
For the entire collection see [Zbl 0895.00045].

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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