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Complex valued systems with an indefinite matrix term arise in importantapplications such as for certain time-harmonic partial differential equationssuch as the Maxwell's equation and for the Helmholtz equation. Complex systemswith symmetric positive definite matrices can be solved readily by rewritingthe complex matrix system in two-by-two block matrix form with real matriceswhich can be efficiently solved by iteration using the preconditioned squareblock (PRESB) preconditioning method and preferably accelerated by theChebyshev method. The appearances of an indefinite matrix term causes howeversome difficulties. To handle this we propose different forms of matrixsplitting methods, with or without any parameters involved. A matrix spectralanalyses is presented followed by extensive numerical comparisons of variousforms of the methods.

Robust iteration methods for complex systems with an indefinite matrix term