A complex Jacobi iterative method for the indefinite Helmholtz equation

An iterative procedure is described for the solution of the indefinite Helmholtz equation that is a two-step generalization of classic Jacobi iteration using complex iteration parameters. The method converges for well-posed problems at a rate dependent only upon the grid size, wavelength and the effective absorption seen by the field. The use of a simple Jacobi preconditioner allows the solution of 3D problems of interest in waveguide optics in reasonable runtimes on a personal computer with memory usage that scales linearly with the number of grid points. Both the iterative method and the preconditioner are fully parallelizable.