Abstract. An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner. 1. Introduction. In this paper we present a novel preconditioner for high wavenumber Helmholtz problems in heterogeneous media. The preconditioner is based on the Helmholtz operator, where an imaginary term is added. This preconditioner can be handled by multigrid. This is somewhat surprising as multigrid, without enhancements, has convergence troubles for the original Helmholtz operator at high wavenumbers.A part of this paper is therefore reserved for the analysis of the multigrid method for Helmholtz problems with a complex zeroth order term. This is done, for constant wavenumbers, by means of Fourier analysis. The preconditioned system leads to a favorably clustered spectrum for a Krylov subspace convergence acceleration. As the preconditioner is not based on a regular splitting of the original Helmholtz problem, it must be used in the setting of Krylov subspace methods. The particular example presented can be viewed as a generalization of the work by Bayliss, Goldstein, and Turkel [3] from the 1980s, where the Laplacian was used as a preconditioner for Helmholtz problems. This work has been generalized by Laird and Giles [17], proposing a Helmholtz preconditioner with a positive sign in front of the Helmholtz term. In [13] we have proposed a preconditioner with a purely imaginary shift added to the Laplacian. The method here is an improvement of that method.In this paper we benefit from Fourier analysis in several ways. First of all, for idealized (homogeneous boundary conditions, constant coefficients) versions of the preconditioned system it is possible to visualize its spectrum for different values of the wavenumber, as Fourier analysis provides all eigenvalues. Second, for analyzing multigrid algorithms quantitatively, Fourier smoothing, two-, and three-grid analysis [6,7,23,24,30] are the tools of choice.

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called "shifted Laplace" preconditioners of the form ∆φ − αk 2 φ with α ∈ C. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.

For various applications, it is well-known that a multi-level, in particular twolevel, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

Abstract. Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm, we are able to provide an optimal value for the shift and to explain the mesh-dependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem. 1. Introduction. In this paper we investigate the spectral behavior of iterative methods applied to the time-harmonic wave equation in heterogeneous media. The underlying equation governs wave propagation and scattering phenomena arising in acoustic problems in many areas, e.g., aeronautics, marine technology, geophysics, and optical problems. In particular, we look for solutions of the Helmholtz equation discretized by using finite difference, finite volume, or finite element discretizations. Since the number of grid points per wavelength should be sufficiently large to result in acceptable solutions, for very high frequencies the discrete problem becomes extremely large, prohibiting the use of direct solution methods. Krylov subspace iterative methods are an interesting alternative. However, Krylov subspace methods are not competitive without a good preconditioner.Finding a suitable preconditioner for the Helmholtz equation is still an area of active research; see, for example, [7]. A class of preconditioners that has recently attracted considerable attention is the class of shifted Laplace preconditioners. Preconditioning of the Helmholtz equation using the Laplace operator without shift was first suggested in [1]. This approach has been enhanced in [8,9] by adding a positive shift to the Laplace operator, resulting in a positive definite preconditioner. In [2,3,4,13] the class of shifted Laplace preconditioners is further generalized by also considering general complex shifts.It is well known that the spectral properties of the preconditioned matrix give important insight in the convergence behavior of the preconditioned Krylov subspace methods. Spectral analyses for the Helmholtz equation preconditioned by a shifted

In this paper we survey the development of fast iterative solvers aimed at solving 2D/3D Helmholtz problems. In the first half of the paper, a survey on some recently developed methods is given. The second half of the paper focuses on the development of the shifted Laplacian preconditioner used to accelerate the convergence of Krylov subspace methods applied to the Helmholtz equation. Numerical examples are given for some difficult problems, which had not been solved iteratively before.