  In this paper, we consider an efficient iterative approach to the solution of
the discrete Helmholtz equation with Dirichlet, Neumann and Sommerfeld-like
boundary conditions based on a compact sixth order approximation scheme and
preconditioned Krylov subspace methodology. A sixth order compact scheme for
the 3D Helmholtz equation with different boundary conditions is developed to
reduce approximation and pollution errors, thereby softening the
point-per-wavelength constraint. The resulting systems of finite-difference
equations are solved by different preconditioned Krylov subspace-based methods.
In the majority of test problems, the preconditioned Generalized Minimal
Residual (GMRES) method is the superior choice, but in the case of sufficiently
fine grids a simple stationary two-level algorithm proposed in this paper in
combination with a lower order approximation preconditioner presents an
efficient alternative to the GMRES method. In the analysis of the lower order
preconditioning developed here, we introduce the term "$k$-th order
preconditioned matrix" in addition to the commonly used "an optimal
preconditioner". The necessity of the new criterion is justified by the fact
that the condition number of the preconditioned matrix $ AA^{-1}_p $ in some of
our test problems improves with the decrease of the grid step size. In a simple
1D case, we are able to prove this analytically. This new parameter could serve
as a guide in the construction of new preconditioners. The lower order direct
preconditioner used in our algorithms is based on a combination of the
separation of variables technique and Fast Fourier Transform (FFT) type
methods. The resulting numerical methods allow efficient implementation on
parallel computers. Numerical results confirm the high efficiency of the
proposed iterative approach.
