Eigenanalysis of Some Preconditioned Helmholtz Problems
Abstract
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.
(Also cross-referenced as UMIACS-TR-98-22)