Field of values analysis of preconditioners for the Helmholtz equation in lossy media
Provides theoretical convergence guarantees for iterative solvers in wave propagation simulations with losses, relevant to computational electromagnetics and acoustics.
The paper analyzes convergence of preconditioned GMRES for Helmholtz equation in lossy media, providing field-of-values bounds for Laplace, inexact Laplace, and two-level preconditioners, with the two-level analysis covering previously unconsidered media types.
In this paper, we analyze the convergence of the preconditioned GMRES method for the first order finite element discretizations of the Helmholtz equation in media with losses. We consider a Laplace preconditioner, an inexact Laplace preconditioner and a two-level preconditioner. Our analysis is based on bounding the field of values of the preconditioned system matrix in the complex plane. The analysis takes the non-normal nature of the linear system naturally into account and allows us to easily consider certain type of inexact Laplace preconditioners via a perturbation argument. For the two-level preconditioner, our convergence analysis takes into account a media, which has not been considered in previous works.