GMRES-based multigrid for the complex scaled preconditoner for the indefinite Helmholtz equation
This work provides a practical solution for solving the indefinite Helmholtz equation, which is important for applications in acoustics and electromagnetics, but the improvement is incremental as it combines existing techniques.
The paper addresses the instability of stationary smoothers in multigrid preconditioners for the indefinite Helmholtz equation by replacing them with GMRES iterations, resulting in a robust and efficient solver for complex shifted Helmholtz problems. The method requires very few GMRES iterations per level and shows good convergence on benchmark problems.
Multigrid preconditioners and solvers for the indefinite Helmholtz equation suffer from non-stability of the stationary smoothers due to the indefinite spectrum of the operator. In this paper we explore GMRES as a replacement for the stationary smoothers of the standard multigrid method. This results in a robust and efficient solver for a complex shifted or stretched Helmholtz problem that can be used as a preconditioner. Very few GMRES iterations are required on each level to build a good multigrid method. The convergence behavior is compared to a theoretically derived stable polynomial smoother. We test this method on some benchmark problems and report on the observed convergence behavior.