Efficient iteration methods for complex systems with an indefinite matrix term

Complex valued systems with an indefinite matrix term arise in important applications such as for certain time-harmonic partial differential equations such as the Maxwell’s equation and for the Helmholtz equation. Complex systems with symmetric positive definite matrices can be solved readily by rewriting the complex matrix system in two-by-two block matrix form with real matrices which can be efficiently solved by iteration using the preconditioned square block (PRESB) preconditioning method and preferably accelerated by the Chebyshev method. The appearances of an indefinite matrix term causes however some difficulties. To handle this we propose different forms of matrix splitting methods, with or without any parameters involved. A matrix spectral analyses is presented followed by extensive numerical comparisons of various forms of the methods.