Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian

Abstract

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

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…This is the same equation of a circle in the complex plane that was found in [36] for the eigenvalue spectrum of the preconditioned Helmholtz equation. Due to this, the following corollary of Theorems 3.…”

“…It is not evident that the results presented in [36] for the Helmholtz equation with C f = 0 are applicable for the Navier equation. However, numerical experiments in Section 6 suggest that similar behaviour to the one described by Theorems 3.4-3.6 in [36] holds also here. The following states this as a conjecture.…”

“…In [36], Theorems 3.1-3.6 give useful information of the eigenvalue spectrum of the preconditioned Helmholtz operator. Some of these results can be generalized to the Navier equation, as will be shown in the following.…”

“…For the analytical study of eigenvalue spectra in Section 4, it is practical to define the following matrices based on the integrals in (13) and (15): [36], the discretized Helmholtz and Navier operators have matrix forms…”

“…This preconditioner will be called a damped Navier preconditioner. Results considering the eigenvalue spectrum of the shifted-Laplacian preconditioned discretized Helmholtz equation were given in [36] and some of these will be generalized to the Navier equation. Simulations are carried out in twodimensional and three-dimensional computational domains including complicated geometries for both Helmholtz and Navier problems.…”