The paper presents a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian, an analysis which is valid under general conditions. Thus, the authors investigate the spectral behaviour of iterative methods applied to the time-harmonic wave equations in heterogeneous media.
The first section represents an introduction concerning the shifted Laplace preconditioners.
The second section describes the acoustic wave equation and its discretization. The authors specify some properties of the matrices (symmetry, complex valued, positive definiteness etc.), which form the coefficient matrix of the resulting linear system.
The third section focuses on spectral analysis of the Helmholtz operator preconditioned with the shifted Laplace preconditioner. The authors show that for the damped Helmholtz equation with Dirichlet and Neumann boundary conditions, the eigenvalues are located on a line or on a circle with a given parametrization for a special type of damping. For radiation boundary conditions and for general viscous media they show that the eigenvalues are located on one side of the line or within the circle.
In the fourth section they combine the results of the spectral analysis presented in the previous section with a upper bound on the GMRES-resiudual norm, which assumes that the spectrum is enclosed by a circle. Using this bound they derive the optimal value of the shift in the shifted Laplacian preconditioner.
In order to illustrate and verify the theoretical results, the fifth section presents numerical experiments performed with MATLAB.