Robust iteration methods for complex systems with an indefinite matrix term

TL;DR

This paper develops and analyzes robust iterative methods for solving complex-valued systems with indefinite matrices, which are common in electromagnetic and wave equations, providing new splitting techniques and spectral analysis.

Contribution

It introduces novel matrix splitting methods tailored for indefinite complex systems, along with spectral analysis and extensive numerical comparisons to demonstrate effectiveness.

Findings

New splitting methods improve convergence for indefinite systems

Spectral analysis guides the choice of iterative schemes

Numerical results show enhanced efficiency over existing methods

Abstract

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.