An efficient preconditioner for iterative solution of the problem is constructed. The problem is discretized with fourth-order accurate finite difference operators. For the construction of the radiation boundary conditions also a fourth-order finite element method is used. The memory requirements for the preconditioner are nearly linear in the number of unknowns. As a model problem electromagnetic wave propagation in a three-layered waveguide is used.
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.