Eigenanalysis of Some Preconditioned Helmholtz Problems

In this work we calculate the eigenvalues obtained by preconditioning the discrete Helmholtz operator with Sommerfeld-like boundary conditions on a rectilinear domain, by a related operator with boundary conditions that permit the use of fast solvers. The main innovation is that the eigenvalues for two and three-dimensional domains can be calculated exactly by solving a set of one-dimensional eigenvalue problems. This permits analysis of quite large problems. For grids fine enough to resolve the solution for a given wave number, preconditioning using Neumann boundary conditions yields eigenvalues that are uniformly bounded, located in the first quadrant, and outside the unit circle. In contrast, Dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased. These eigenvalue properties yield the first insight into the behavior of iterative methods such as GMRES applied to these preconditioned problems.