The problem is discretized with fourth-order accurate finite difference operators. For the construction of the radiation boundary conditions a fourth-order finite element method also is used. The memory requirements for the preconditioner are nearly linear in the number of unknowns. The arithmetic complexity for each iteration is low, whereas the construction of the preconditioner is a bit more expensive. Electromagnetic wave propagation in a three-layered waveguide is used as a model problem.
Numerical experiments show that convergence is achieved in a few iterations.
Compared with banded Gaussian elimination, which is a standard solution method, the iterative method shows significant gain in both memory requirements and arithmetic complexity. Furthermore, the relative gain grows when the problem size increases. Helmholtz equation, domain decomposition, finite difference discretization, preconditioned iterative method, fast transform AMS Subject Classifications . 65F10, 65N22 PII .