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A complex Jacobi iterative method for the indefinite Helmholtz equation. (English) Zbl 1069.65110

Summary: An iterative procedure is described for the solution of the indefinite Helmholtz equation that is a two-step generalization of the classical 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.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65Y05 Parallel numerical computation
78A50 Antennas, waveguides in optics and electromagnetic theory
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References:

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