Field of values analysis of a two-level preconditioner for the Helmholtz equation (Q2845603)

The paper presents a convergence study of the generalized minimal residual (GMRES) method applied to the first-order finite element discretisation of the Helmholtz equation preconditioned by a two-level multi-grid preconditioner. The matrices arising from the discretisation are complex, non-Hermitian, and non-normal. Due to the non-normality, the analysis based on the eigenvalues of the preconditioned matrix may not provide meaningful convergence bounds. In the paper, the convergence analysis is based on the field of values; in particular, the bounds on the location of the field of values in the complex plane are derived in terms of the wave number, loss term, and coarse and fine grid mesh sizes of the two-level method. As the result, the convergence of the GMRES method is proved to be independent of the wave number \(\kappa\) and the mesh sizes provided that the coarse grid mesh size \(H\) satisfies the constraint \(\kappa H^{3}\ll 1\). The theoretical findings are illustrated on 2- and 3-dimensional examples.