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Advances in iterative methods and preconditioners for the Helmholtz equation. (English) Zbl 1158.65078
The Helmholtz equation $$\nabla ^{2}u(\mathbf{x})+\kappa ^{2}u(\mathbf{x})=h( \mathbf{x}),$$ where $$\nabla ^{2}$$ is the Laplacian, $$\kappa$$ is the wave number, $$h$$ is a forcing function and $$u$$ is the amplitude, finds applications in many important fields including aeroacoustics, under-water acoustics, seismic inversion and electromagnetics. Therefore computation of its solutions in a two or three dimensional domain is important. The linear system arising from a discretization of the Helmholtz equation is typically characterized by indefiniteness of the (real part of the) eigenvalues of the corresponding coefficient matrix, and hence the corresponding iterative scheme may encounter convergence problems.
This paper reviews and highlights some recent advances in iterative methods for the Helmholtz equation. In particular, the author focuses on the Krylov subspace methods and the shifted Laplacian preconditioner. Some theories behind the shifted Laplacian preconditioner are given, and numerical results are presented for realistic problems. There are 142 references listed in this paper which include several recent surveys on similar subjects. The emphasis is on engineering computation, however, and the reader who is interested in the theoretical aspects is therefore encouraged to look for additional sources as well.

##### MSC:
 65N06 Finite difference methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F35 Numerical computation of matrix norms, conditioning, scaling 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
##### Software:
symrcm; PARDISO; Wesseling; LFA; ILUT; CGS
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